In Pittsburgh the new Westinghouse controlled cars can run every two interfering traffic. transit expressway shows how com- minutes, day and night. They operate The Westinghouse transit express- muting can be made comfortable, con- singly when passenger load is light, way costs less to install and operate venient, fast and fun. form trains to handle heavy traffic. than a conventional subway or elevated. A two-mile pilot line has been built Cars are fast and safe. They run si- If your city had one, you would to demonstrate the feasibility of this lently and smoothly on rubber tires probably never find it necessary to new system. Lightweight, computer- along their own expressway, free of drive to work again. Does this look like an answer to global problems of ignorance, disease and physical deprivation? Not yet. But we're getting closer. collect it, classify it, store it...and distribute it appropriately and instantly to the people who need it. Behind this movement is a simple statistic with startling im- plications for all of us — If you count all the scientists and In this light, you might consider today's Xerox products early engineers since the beginning of recorded history, ninety and primitive steps along a difficult but fascinating path. percent are alive today! You'd be right. Yet, has anyone taken these steps before us? We are now in the midst of the result — an incredible explo- Your degree in Engineering, Science, Business Administration sion of information from every corner of the globe. And or Liberal Arts can qualify you for some intriguing openings somewhere within this explosion will be the ultimate answers at Xerox, in fundamental and applied research, engineering, to mankind's oldest, and newest problems. manufacturing, marketing/sales, finance and administration. See your Placement Director or write directly to Mr. Stephen G. The challenges are many. First, to understand the nature Crawford, Xerox Corporation, P.O. Box 1540, Rochester, New of this giant intellectual force. Then, to find the best way to York 14603. An Equal Opportunity Employer. This is Olin. Who is Olin? Olin is a world-wide company with 39,000 employees developing, producing and marketing products from eight divisions: Squibb, Winchester-Western, Chemicals, Metals, Agricultural, Ecusta, Film, and Forest Products. With corporate offices in New York City, the firm operates 60 plants in 30 states with plants and affiliates in 37 foreign countries. What are the types of work at Olin? Olin's great diversity provides a broad range of opportunities in the technical science and engineering fields. Emphasis is placed on the B.S. and M.S. chemical, industrial, mechanical and metallurgical engineering student for assignments in plant operations, process control, product development, quality control, production and marketing. Advanced degree M.S. and Ph.D. chemists and metallurgists work in central research and development improving existing products and developing new ones. Men with liberal arts and business backgrounds find rewarding career opportunities in the administrative functions, marketing, and some areas of manufacturing. What are the opportunities at Olin? Olin recognizes people as its greatest asset. Your future growth and career is as important to the company as it is to you. You will be given thorough on-the-job training in your first job to prepare you to accomplish your career objective. You will learn and progress, according to your ability, working with skilled and experienced men in various assignments. For additional information about Olin, please contact your Placement Office or write Mr. Monte H. Jacoby, College Relations Officer, Olin, 460 Park Avenue, New York, N.Y. 10022. FRED GEORGE . . . . . . . . . . . . . . . . . editor ARTICLES TOM McCLURE . artist OUR NEW PROGRAMS - WHY An explanation of the new look in engineering at MSU Dean J . D. Ryder 9 staff JOE BOWMAN BIOLOGICAL RECOVERY OF WASTE WATER ANDREW POOLE Dr. Karl L. Schulze 13 RICHARD MURRAY DEE GOSSETT FUN WITH SQUARES Some arithmetic shortcuts 27 SUSAN GOODSELL BRIAN GOLUSKA EXTENSION OF THE PYTHAGOREAN GARY ROMANS THEOREM TO N-DIMENSIONS T. Heppenheimer 36 MARIAN LOCKE HOMECOMING DISPLAY Thekid 45 advisors J , RYDER DEPARTMENTS C. MENSENDICK Editorial 4 T.FARRELL Placement Bureau 6 Industrial Spotlight 18 Industrial News 42 Engineers 46 Advertisers" Index 47 Editorial.. THE DRAFT Since July of this year the quotas to be filled by the Selective Service System of this Country have been rising at an ever-increasing rate. We have come from a call of 1,700 in February and words of abolishing the Draft to a call of over 45,000 in December. This can be easily understood in the context of our rising participation in the war in Vietnam. However, what is so difficult to comprehend is why President Johnson and the Se- lective Service have decided to threaten the college student with induction into the Armed Forces. This is not meant to disparage the current administration's foreign policy, or the Selective Service itself, or defending one's country through service in the Armed Forces of the United States. Indeed, the service does have its time and place — after graduation. We have often been graced with such glowing names as the "Hope of Tomorrow" and the "Leaders of Tomorrow" from people occupying im- portant positions within the government. Why don't they treat us as they speak of us? They don't show any thought. They are willing to take us right out of the classroom for induction. Aren't they aware that to be leaders we must be educated? Don't they know that only a small percentage of those who interrupt their education by two years or more, return to school? Let us finish our education first. We will be more valuable to our Country as well as ourselves. Another of your future's many facets at Monsanto As a billion-dollar manufacturer of hundreds of products—in 40 U. S. plants, 15 overseas—Monsanto considers the manufacturing engineer of prime importance. His talents are needed to find new ways (and improve old ways) of increasing yields, improving control, lowering costs—bringing plants up to optimum performance. You will have the satisfaction of applying your engineering skills and knowledge, and seeing the results . . . from inception to final operation . . . and have the full responsibility for your decisions. There are production challenges now awaiting good men . . . plus substantial rewards for meeting them. Let us show you what they can mean to you . . . professionally, personally, financially. Write today—we'll send you our brochure, "Your Future and Monsanto." Address: Manager, Professional R e c r u i t i n g , D e p t . 9 6 2 , Monsanto, S t . Louis, Missouri 63166. An Equal Opportunity Employer November 22 January 13 Institute of Paper Chemistry Pittsburgh Plate Glass Chem. and Chem. Eng. Carnation Company Packaging Corp. of America The Udylite Corporation Ch., Ch. E., M.E., E.E. January 17 Continental Can Co. Sylvania M.E. Continental Can Corp. American Bosch Arma Republic Steel Corp. All Eng. January 18 Leeds & Northrup Co. November 23 Ohio Edison U.S. Army Tank-Automotive Center Pennsalt Chemicals M.E. Allegany Ballistics Laboratory Interlake Steel Corp. Brown Engineering Corp., Inc. All Eng. January 19 Vickers Inc. Division, Sperry Rand B. F. Goodrich Corp. Union Carbide M.E., E.E. Deering Milliken Service Corp. January 20 Ch.E., E.E., M.E. Prestolite Company Armstrong Cork Company November 29 General Telephone and Electronics Ohio Department of Highways Laboratory C.E. January 21 U.S. Army Engineers Waterways Wyandotte Chemical Corp. Experimental Station United Aircraft E.E., E.E., Physics December 2 January 25 Sundstrand Corp. General Radio Co. M.E., E.E. Consumers Power Fairbanks Morse Inc. U.S. Naval Avionics Facility E.E. January 27 December 3 Allied Chemical Corp. Standard Brands, Inc. M.E., Ch.E., Ind. E., Chem., January 31 and February 1 Biochem. North American Aviation Inc. Carve fast and with the flourish recent development is graphite Whose new of a master chef whether you are in the woods of Maine, your own textiles used both in the white heat of rocket blasts and the ex- battery lets back yard, or having a formal din- ner party. treme high temperatures of in- dustrial furnaces. And compact fuel cells, which generate electric you carve like New cordless electric carving knives take full advantage of bat- tery power. They are conveniently power by a chemical reaction and provide a whole new source of an expert portable and weigh half as much as leading cord models. What's e n e r g y , are also now b e i n g marketed. anywhere? more, these knives are so power packed they can slice roast beef, ham, turkey and other meats in a To keep bringing you these and many other new and improved products, we'll be spending half a jiffy, making neat slices all the billion dollars on new plant con- way. Their energy comes from struction during the next two a new kind of EVEREADY nickel- years cadmium battery developed by Union Carbide. It's rechargeable. A lot of new things are hap- pening at Union Carbide. Another If you still think glass is just glass, ask NASA. Ask NASA about the windows on the Gemini spacecraft Conduct or insulate. Bend. Not bend. Break. Not break. —formed from glass specially designed by Corning. Melt. Not melt. Glass can be made to do what you want They not only provide mechanical and thermal protec- it to. It is the most versatile basic engineering material. tion, but also increase visual transmission beyond the For solutions to their problems, government and in- theoretical 92 per cent to 99 per cent. Ask, too, about the dustry are coming to Corning. Because Corning is the high-speed glass memories that help guide Gemini to in- space rendezvous and re-entry. glass-master. It's a broad, international company, with one of the most daring, expert and imaginative research and And ask about the glass resistors, as thick as telephone poles, which serve as dummy loads and power dissipating engineering staffs in the world. Plus, a marketing prin- terminations at tracking stations. ciple that commits the company to developing products only in areas where a need exists and no product does. Glass can be designed to maintain constant electrical Young engineers seeking challenge, opportunity, and properties at missile speeds. Withstand the pressures at the ocean's floor. Save weight without sacrificing strength. advancement are invited to write to Corning's Career De- velopment Manager. Our New Programs- Why It has been said that engineering out of the laboratory in 1948 and education by the advancement in knowledge is growing at the rate of was in application by 1953. all technical fields are having a about ten percent per year. Obvi- Each of these inventions has beneficial effect in requiring study ously, it is not possible to expand called for the development of a of the engineering-education pro- the engineering curriculum at a complete new academic discipline, cess and its basic philosophy. Our similar rate and provide the sen- frequently requiring tremendous progress is a result of such pres- iors with a curriculum which is amounts of research to provide sures. forty-five percent larger than the the fundamental knowledge which Students at Michigan State Uni- one in existence when they became must underly the application. For versity are immersed in an atmos- freshmen. This rate of technologi- instance, it has been said that in phere of dynamic education of cal growth is posing tremendous 1948 germanium was one of the many types. An engineering stu- questions for all the engineering lesser-understood chemical ele- dent, associating with other stu- colleges. ments. Because of its importance dents on the campus, cannot avoid In contrast, during the cycle of to the transistor and because of becoming aware of the educational development, refinement, and util- the huge economic potential of that philosophies and objectives of stu- ization of reciprocating steam en- device, it is now believed that we dents projecting their futures into gines there was ample time, in know more about germanium than many diverse fields. The College fact several hundred years, for we do about any of our other chem- of Engineering cannot remain aloof training of engineers in the de- ical elements. and point its program differently; velopment and application of this What does this mean with r e - it knows that the objective of uni- aspect of technology. Today we spect to the program which our versity education should be the have seen the rocket motor, which students are today undertaking in preparation of a young man or was not much more than a figment the College of Engineering? We young woman for forty years of a of Goddard's imagination in 1923, c a n n o t continue to expand our productive, useful, and enjoyable phenomenally developed in the last program in step with the advance life. Engineering must look beyond twenty years to the two and one- of new technology because of eco- the preparation of its graduates half million horsepower level of nomic and social limits on the pe- for immediate usefulness in a field Saturn V. Radar was developed riod of time which our society can of technology to the usefulness of from a mere idea in 1932 to the justify in the education of its that graduate for his whole life- device that won the Battle of Brit- younger generation. However, the time as well. Thus the teaching of ain in 1940. The transistor came pressures placed on engineering a technology or application, which may become obsolete a few years grams from a former high level of ble loss on the technical side. At after graduation, should be chal- credit hour requirements to the the same time, the number of hours lenged by a student as Inefficient level which approximates the r e - devoted to free electives has been use of his time on this campus. quirements in the rest of the Uni- increased, and because of the gen- versity. Our step has been cheered eral l o w e r i n g of requirements Our Engineering College and by industry and other engineering from the eighteen-nineteen credit others have been searching for the educators as well. Soon after our level per term to the fourteen- common denominator which will reduction was announced, two uni- sixteen credit level, there exists lead to selection of appropriate versities which had been conduct- many opportunities for the student curricular material which can be of ing five-year Bachelor of Science to sample additional courses of- benefit over a student's profes- p r o g r a m s in engineering, an- fered elsewhere on the campus, sional lifetime. We have attempted nounced that they were returning without taking on heavy overloads. to find that material in mathe- to the more conventional four-year All of this reflects the desire matics and the basic and engineer- programs. Several other schools of this College for its engineering ing sciences, since it seems prob- have reduced their credit hour r e - g r a d u a t e s to be t h o r o u g h l y able that Newton's laws of motion quirements also and this indicates grounded in that knowledge which will be more useful ten years after that our change was in the direc- we believe will be of value over the graduation than will be knowledge tion of the trend. next forty years. We also desire of the test code for an engine or that our graduates be well-rounded motor. In addition we have recog- The reduction in required hours citizens capable of applying their nized that the scope of engineering reflected the philosophy that over engineering abilities in both the education is now so broad that to some years the College had ceased technical and social areas. This specialize a student in one aspect requiring various courses in en- means that the days are past when may be to deny him opportunity in gineering applications, and that it an engineering specialist could other more challenging areas later was time the credit hours were withdraw into his own realm and in his lifetime. Thus we find the taken out as well. We also desired become merely a servant of tech- emphasis on mathematics as a tool that engineering students have ev- nology. Rather, the engineer now and a language, the engineering ery opportunity for extra-curricu- will be called upon and must be sciences, and engineering methods lar activity, and for broadening of capable and willing to apply his which will furnish the raw materi- t h e i r programs through self- abilities in solving not only the als for ideas, concepts, and de- chosen electives, that are avail- technical problems, but also those signs for many years after grad- able to students in other Colleges which in the past have been viewed uation. on this campus. Through increased as in the political, economic, or efficiency in the teaching of the social realm as well. Added to the above is the transi- necessary areas in engineering, we tion which students now in the have been able to carry out the r e - College of Engineering are wit- duction and introduce new upper nessing in the change of our pro- school programs without apprecia- You can join IBM. And help shape computers are only tools—they For more information or if you a better world. can't think. And that's where you come missed our interviewers, write to: The key is IBM computers. They're in. In your hands, man's most daring Manager of College Relations, helping medical researchers learn dreams can become reality—to make IBM Corporate Headquarters, Armonk, more about cancer and factors a better life for more people. New York 10504. IBM is an Equal associated with it. Helping educators If that sounds like an exciting Opportunity Employer. develop new techniques for instructing prospect, it's because it is. A career culturally deprived children. Helping with IBM will give you both personal engineers plan flood-prevention and professional fulfillment. programs. Your placement officer can tell you Computers are helping to do a lot of about career opportunities with IBM. good things for a lot of people. But See him, too, for an appointment with our interviewers. Before you conquer new worlds, try your hand at reshaping the old one. MEET THE CLASS OF '65 They're members of Bethlehem Steel's 1965 Loop being assigned. Course — graduates of colleges and universities Most Loopers Have Technical Degrees from coast to coast. Our primary need is for engineering and other What is the Loop Course? Since 1922, Bethlehem technical graduates. Such men have many fine has conducted this Course for training college opportunities in all phases of steelmaking, as well graduates for careers in Bethlehem's management. as in research, sales, mining, fabricated steel con- Hundreds of men at all levels of Bethlehem man- struction, and shipbuilding. Also, both technical agement, including our Chairman, started their and non-technical graduates are needed for other careers as loopers. important activities including accounting, sales, The '65 Loop convened at our general offices in purchasing, traffic, finance and law, industrial and Bethlehem, Pa., early in July. After five weeks of public relations, and general services. thorough indoctrination, many of these men were You'll find a great deal more information in our assigned to facilities throughout the country for booklet, "Careers with Bethlehem Steel and the further brief training at the operations before Loop Course." You can obtain a copy at your undertaking their first job assignments. Others, Placement Office, or drop a postcard to Personnel such as sales and accounting trainees, may remain Division, Industrial and Public Relations Depart- at the general offices for some months before ment, Bethlehem, Pa. 18016. BIOLOGICAL RECOVERY OF WASTE WATER The quality of our surface wa- formed in the aeration tanks. The waste water. The first set of tanks ters, rivers, and lakes, and even overflow from the final settling consists of three glass aquari- that of our ground water, is de- tanks is called Final Effluent, and ums. These were started in May, creasing from year to year. Fish is discharged over two cascades 1964, and have been in continuous kills occur in the Red Cedar River, into the Red Cedar River. This operation for the one and a half the river which winds its way type of treatment is called the Ac- years since. The first aquarium through the campus of Michigan tivated Sludge process. It repre- has a fifteen gallon capacity, and State University. Swimming beach- sents our most complete treatment is equipped with an arrangement es on Lake Erie, Lake Huron, and process to date, and is capable of of fibre-glass screen, three air- Lake Michigan, have to be closed removing eighty to ninety per cent diffusers, airflow meter, thermo- every year. of the pollutional material con- meter, dissolved oxygen metering There is general agreement that tained in the original raw sewage. electrode, and three fluorescent pollution by municipal and in- Actually, the East Lansing plant lamps. The lamps are operated by a dustrial wastes is the main cause is too small to handle the waste timer so that they are on fifteen of this deterioration. If it be- load properly, and the efficiency, hours a day, and off nine hours a comes necessary to put a pri- therefore, is considerly lower than day thus imitating day and night that. The same problem arises in conditions. The fiberglass mesh- ority on any phase of Michigan many a town where the population work is the characteristic feature water problems, it now appears is rapidly increasing year by year. of the tank, and it is this that he that Water Pollution should have believes to be the most active part that priority. By polluting and This old plant will be abandoned shortly and a new plant will take of the system. The pieces of screen defiling the sources of our water are mounted vertically, and par- supplies, we are sowing the seeds over. allel to each other at half inch of national destruction. We are Anyway, normally ten to twenty distances, somewhat similar to the creating a crisis by our self- percent of the incoming pollutional way the plates in a car battery seeking short-sightedness. load is placed as a burden on the are arranged. What do we actually do with our river. Due to the ever-increasing municipal waste water, for in- population we have just mentioned, This provides a large amount stance? Let us use our own cam- this burden will inevitably like- of surface for the development of pus and the city of East Lansing wise increase and there is no doubt a fascinating community of small as examples. The sewage from that in the future such so-called organisms which is called, "Epl- these two sources amounts to about secondary waste treatment meth- phyton." It consists mostly of five million gallons per day, and ods will not be sufficient any more. bacteria, diatoms, protozoa, roti- is treated at the East Lansing Tertiary, or third stage treat- fers, and worms. Apparently this s e w a g e treatment plant. After ment methods will have to be de- biological community has a r e - passing through a bar screen, the veloped with the aim of remov- markable capacity for cleaning raw waste water is pumped into ing the many different types of up polluted water. four primary settling tanks, where w a t e r contaminents to a much In fact, our data show that with the heavier solids sink to the bot- higher degree than we are now this tertiary treatment method it tom, and are then pumped out as practicing, and this is where Dr. is possible to remove 85 to 95 s l u d g e . This form of primary Schulze's work begins. In 1964 he per cent of the pollutional load, treatment is capable of removing started a project to which he gave which has remained after the sec- about thrity-five per cent of the the title, "Biological Recovery of ondary treatment by the plant out- incoming pollutional load. Waste Water." side this building. To check on the The next treatment step con- Final effluent from the East Lan- quality of the tertiary treatment sists of ten aeration tanks where sing plant is used as the raw ma- effluent a second tank was installed the major part of the polluting terial. A small centrifugal pump behind this tank. material is oxidized by the activity delivers a constant flow of the pre- This second tank contains a bot- of bacteria, in the presence of air. treated sewage into a supply tank tom layer of about two inches of The air is blown into the liquid in the River Laboratory Building gravel, several species of aquatic by two large compressors. The ef- to the South of the East Lansing plants collected from Lake Lan- fluent from these aeration cham- Sewage Plant. From the supply sing and the Red Cedar River, and bers is channeled into four final tank, three variable speed pumps two pairs of guppies. The idea was settling tanks, to remove those feed the experimental tanks with simply to find out whether such a suspended solids which have been a continuous flow of pre-treated tertiary treatment would be effec- tive enough to maintain normal of my tertiary treatment process. e x c e l l e n t shape, although Dr. aquatic life in the treated sewage, After these encouraging results Schulze has a feeling that the large- on a long-term basis. So far, our from the small aquariums, he be- mouth bass is pretty hungry most experience has shown this to be came a little more ambitious this of the time. the case. Originally, one pair of summer and increased the scale In general, Dr. Schulze would guppies was placed in the tank, of the experiment from ten to say that the biological recovery and they produced offspring which fifteen gallons to 200 gallons. of waste water has been demon- grew up, and they in turn produced This new tank corresponds to strated as a reality by these ex- offspring, and now this is the third the previous reactor tank except periments, and he would like to generation of guppies, and they are that it is larger and that two emphasize that it is nothing but about to produce their young fish blocks of plastic foam are used treated sewage that flows through in the next month. Dr. Schulze instead of fiberglass screens. This these tanks. There is no dilution would like to stress that these new tank has now been in opera- by tap water, river water, or lake fish have never received any out- tion for about five weeks, and it water, being used in the process. side food. They exist entirely on is not only producing an excellent the living food produced in this effluent, but it is harboring some He has been asked such a type tank. In fact, the aquarium has intersting fish life. Two blocks of tertiary treatment could be de- maintained itself in an excellent of plastic foam create three com- veloped for large scale use on a condition over the eighteen months partments in this tank. State-wide or even a Nation-wide since it was connected to Tank One. basis. He is certainly going to try In compartment one, he placed to develop this process through a The only thing that has to be done two g o l d f i s h and compartment is remove the excess of young gup- s e r i e s of pilot plants until it three was stocked with one shiner, reaches a stage where its prac- pies and the excess of plant growth. one dace and one bluegill. In other Every three weeks relatively large tical possibilities can be fairly words instead of guppies we are well judged. So far the project amounts of Potamogeton, Cerato- now using representatives of Mich- phyllum, Elodea, and Vallisneria has been supported only on a very igan fish life. This was done about limited scale by several small have to be taken out to prevent four weeks ago. The results are crowding. grants from the Institute of Water surprisingly good. All fish are do- Research. It is hoped, however, This reminds us of an intrigu- ing very well, in spite of the fact that substantial federal grants can ing side-aspect of Dr. Schulze's that they do not receive any out- be obtained in the future. Next, project. The U.S. Department of side food and have to live in noth- the interested v i s i t o r usually Defense maintains a research lab- ing but treated waste water. In brings up the economical view- oratory at the Brooks Air Force fact the goldfish in the first com- point by stating that on a large Base in Texas, where a group of partment are growing so fast that scale such a process would cost scientists is studying ways to grow this indicates an abundance of food a lot of money. This is certainly Algae and Lemna Minor on space being produced. The effluent from true. But, if we have enough money vehicles to produce a food supply Tank A discharged into a second to shoot people to the moon, he feels for astronauts. Lemna Minor is tank, B of the same size. This tank we also should have enough money the botanical name for a small corresponds to the small tank Two, to solve our water problems. water plant known as Duckweed. and like Tank Two it has a gravel It so happens that this Duckweed bottom with a good stand of aquatic As you will probably realize grows in the tertiary treatment plants in it. However, instead of there are many problems con- tank to such an extent that it is guppies, a large-mouth Bass, two nected with this project which necessary to remove half of the Minnows, and a Bluegill were put can only be solved in coopera- growth every two weeks. Since it into the tank. These fish were p r o - tion between several scientific dis- is being considered as food for a s - vided by the Department of Fish- c i p 1 i n e s such as microbiology, tronauts, Dr. Schulze felt that he eries and Wildlife here at Mich- zoology, botany, chemistry and should try it at least once. He igan State University, a depart- physiology. This will become clear washed a handful of it in tap wa- ment which has shown a wonderful by just mentioning the fact that ter and ate it raw. To his surprise cooperative interest in this proj- we have here a continuous flow it had a very pleasant mild taste, ect. Again no outside food is given culture of the whole food chain similar to lettuce. He said that he these fish. from bacteria to protozoa to algae would not mind eating it regularly Just as all the other fish, they to crustaceans to higher plants to as a salad. Anyway, he never ex- must exist on whatever food or- snails and to fish. pected that this potential food for ganisms they can find in their This opens up the possibility astronauts would be a by-product environment. So far they are in of investigating the long term ef- fects of small residual concentra- tions of detergents or pesticides or metals etc., on all these forms of life. There is no doubt in my mind that the approach shown here, combining biology and engineering, will lead in time to a complete recovery of our waste water and to a drastic reduction of the pol- lutional l o a d on our w a t e r re- sources. Dr. Schulze can foresee the time when all our waste treat- ment plants will be implemented by a series of ponds producing edible fish and a sparkling clear effluent of clean reusable water. (Lt. McKinley holds degrees in elec- Air Force research are often as impor- the right places, keeping America strong tronics and electrical engineering from tant, scientifically, as the main thrust. and free. the Georgia Institute of Technology and the Armed Forces Institute of Tech- How important is the work a recent What's the best way to start nology. He received the 1963 Air Force graduate can expect to do? an Air Force career? Research & Development Award for his It's just as important and exciting as his An excellent way—the way I started—is work with inertial guidance compo- own knowledge and skill can make it. through Air Force Officer Training nents. Here he answers some frequently- From my own experience, I can say School. OTS is a three-month course, asked questions about the place of that right from the start I was doing given at Lackland Air Force Base, near college-trained men and women in the vital, absorbing research. That's one of San Antonio, Texas, that's open to both U.S. Air Force.) the things that's so good about an Air men and women. You can apply when Force career—it gives young people the you're within 210 days of graduation, or Is Air Force research really chance to do meaningful work in the after you've received your degree. advanced, compared to what areas that really interest them. others are doing? How long will I be committed What non-scientific jobs does the to serve? It certainly is. As a matter of fact, much Air Force offer? of the work being done right now in uni- Four years from the time you graduate versities and industry had its beginnings Of course the Air Force has a continu- from OTS and receive your commis- in Air Force research and development ing need for rated officers—pilots and sion. If you go on to pilot or navigator projects. After all, when you're involved navigators. There are also many varied training, the four years starts when in the development of guidance systems and challenging administrative-manage- you're awarded your wings. for space vehicles—a current Air Force rial positions. Remember, the Air Force project in America's space program — is a vast and complex organization. It Are there other ways to become you're working on the frontiers of takes a great many different kinds of an Air Force officer? knowledge. people to keep it running. But there are two uniform criteria: you've got to be There's Air Force ROTC, active at intelligent, and you've got to be willing many colleges and universities, and the What areas do Air Force scientists to work hard. Air Force Academy, where admission get involved in? is by examination and Congressional What sort of future do I have appointment. If you'd like more infor- Practically any you can name. Of course in the Air Force? mation on any Air Force program, you the principal aim of Air Force research can get it from the Professor of Aero- is to expand our aerospace capability. Just as big as you want to make it. In space Studies (if there's one on your But in carrying out this general purpose, the Air Force, talent has a way of com- campus) or from an Air Force recruiter. individual projects explore an extremely ing to the top. It has to be that way, if wide range of topics. "Side effects" of we're going to have the best people in United States Air Force Ford Motor Company is: challenge At many companies the opportunity to work on challenging projects comes after many years of apprenticeship and a few grey hairs. Not so at Ford Motor Company where your twenties can be a stimulating period. There are opportunities to prove your worth early in your career. Dale Anderson's experience is a case in point. After receiving his B.A. in Physics in June, 1962, Dale joined our College Graduate Program and was assigned to our Research Laboratories. Recently he was given the responsibility for cor- recting cab vibration occurring on a particular type of truck. His studies showed that tire eccen- tricity was the cause of the trouble. Since little change could be effected in tire compliance, his solution lay in redesigning the suspension system. Tests of this experimental system show the problem to be reduced to an insignificant level. That's typical of the kind of meaningful assignments given to employes while still in the College Graduate Program—regardless of their career interest. No "make work" superficial jobs. And, besides offering the oppor- tunity to work on important problems demanding fresh solutions, we offer good salaries, a highly professional atmosphere and the proximity to leading universities. Discover the rewarding opportunity Ford Motor Company may have for you. How? Simply schedule an interview with our representative when he visits your campus. Let your twenties be a challenging and rewarding time- MOVEABLE WINGS. The Air Force-Navy F i l l supersonic tactical fighter, being developed America's newest hardware byGeneral Dynamics/Fort Worth, borrows another principle from thebirds.Wingsfold backforhigh rolls on Timken bearings speeds, extend perpendicular to the fuselage for slow speeds, maximum lift. The nose wheel and main landing wheels hit the runway on Timken" bearings. THE SHERIDAN. This new XM-551 TAPERED DESIGN. In 1899, THE MARINES ARE COMING! This new LVW landing developmental U.S. Army assault ve- Henry Timken patented the carrier clips along at 35 mph on land or sea. It has hicle can fire conventional rounds or t a p e r e d roller b e a r i n g . four retractable wheels and a 1,500-horsepower gas a Shillelagh through its 152 mm com- Though improved many turbine engine. It weighs 14 tons, rolls trouble-free on bination launcher. Timken tapered times, its basic design is little 34 Timken tapered roller bearings. roller bearings keep the vehicle's changed. Reason: it's the one The Timken Roller Bearing Company, Canton, Ohio. track wheels gung-ho. bearing design that can take Also makers of Fine Alloy Steel and Rock Bits. crushing radial and thrust loads in any combination. In giant military machines that tumble over rough terrain, this fact is all important. OLDSMOBILE TORONADO F r o n t - w h e e l drive d e s i g n , splits the torque between two front duction. Primary stages are 22% though not a new concept, has drive axles. smaller, using a triple venturi for been improved and is being uti- Its power plant, called the Tor- finer, more stable mixture control lized in the Oldsmobile Toronado, onado V-8, is a modified, more providing improved idling and in- called the most unique American powerful version of Oldsmobile's creased economy. Secondaries are automobile in many years. 425 cubic inch engine, and is rated 44% larger, also utilizing more To be produced as a6-passenger at 385 horsepower. Performance precise fuel control called 'air hardtop coupe, the Toronado fea- increases result from improved valve metering' which insures cor- tures unusual interior space for engine breathing and new, more ef- rect air-fuel ratios during accel- passengers and luggage, outstand- ficient carburetion. eration and high performance op- ing driving traction and handling A 4 - b a r r e l carburetor, t h e eration. The new, more simplified c h a r a c t e r i s t i c s , and distinctive Quadrajet, developed for Oldsmo- design uses a single fuel reser- styling. It is 211 inches long, 52.8 bile by Rochester Products divi- voir with just one inlet needle and inches high, and is built on a 119- sion, provides increased air in- inch wheelbase. Toronado design objectives have been achieved through the elimina- tion of the transmission hump and driveline tunnel, the development of an advanced torsion bar suspen- sion system, and the location of all power train components under the hood, well forward of the passenger compartment. The engine is situated as usual fore and aft under the hood, but is offset slightly to the right of center. The transmission's torque con- verter is attached conventionally to the rear of the engine. The r e - maining section of the transmis- sion is turned around and mounted along the left side of the engine, facing forward. A link chain as- sembly transfers power from the converter across and through the transmission gear sets to a dif- ferential unit bolted to the front of the transmission, which in turn How to join DuPont closest to the top Join us right out of school. Few men start higher at Du Pont — or with more promise —than graduating engineers. That's because we fill virtually all important positions from within. It's a policy that's helped us grow 750% since 1937. Since 1950 our sales have more than doubled from $1 billion to $2.4 billion. We spend $90 million a year on research, $60 million of which goes into the discovery of new scientific truths and new materials. And because new products continually emerge from our re- search labs, responsible new career opportunities constantly spring up for our technical men. In fact, there's room for you to actually "change jobs" within Du P o n t - t o seek out the spot best suited to you without loss of your accrued benefits. Consider us if you're ready to make your career decision. You may be closer to the top than you think! Write us a letter, or mark and mail our coupon today! torque is split between two front drive axles; one connected to the planetary gears, the other to an internal sun gear. While the car is moving straight ahead, gears are fixed and rotate at the same speed, but when turn- ing, the planetary gears revolve with differential action, allowing the drive axles to rotate at dif- ferent speeds. Torque is fed from the dif- ferential directly to the left drive shaft, and through a cross shaft under the engine oil pan over to the right drive shaft. Each axle- drive shaft has inboard and out- board constant velocity universal joints so the shafts have upward, downward, forward and rearward freedom required for independent front suspension. Front suspension for the Tor- onado is of torsion bar design. Two specially hardened steel torsion one float to assure a constant fuel tates counter-clockwise,transfer- rods work from each lower con- supply in any driving situation. ring torque from a drive sprocket trol are to special anchors mounted Overall engine operation is im- at the rear of the converter to a in a cross support near the middle proved through a new choke which driven sprocket at the rear of the of the car. The cross support, is mounted in the intake manifold gear sets. mounted with rubber cushions to where it can better sense engine Instead of a long drive shaft the frame, contains single bolt temperatures. running back the entire length of anchor adjustments whereby the A large, low profile, dual snorkel the car underbody, a simple spline carrying height can be adjusted air cleaner with a resin treated shaft connects transmission gear by increasing or decreasing torque filtering element supplies air to the output with the differenetial. Eight on the torsion rod. carburetor. In addition to furnish- bolts fasten the differential di- Both upper and lower control ing air, the tapered snorkels also rectly to the front of the transmis- arms are heavy stampings and contribute to the suppression of sion. specially calibrated shpck absorb- noise, while their efficient, thin The Toronado's differential has ers for fast reacting stability are design allows for a lower hood. been specifically designed for this mounted at an angle. Wheel caster Toronado engine output is in- car. It uses a planetary gear set and camber can be easily adjusted creased by larger intake valves, rather than the standard ring gear with an eccentric cam assembly an increased throat dimension in pinion, allowing for its slender, in the upper control arm, rather the head, and larger, less restric- more compact design. Differential than by the conventional shim tive intake manifold branch areas; method. also, a high-life profile camshaft and larger valve lifters. Teamed with the Toronado V-8 engine as standard equipment is the specially adapted version of Oldsmobile's Turbo Hydra-Matic transmission. In function and con- trols, it is basically the same as the standard three-speed Turbo Hydra-Matic, but in construction and power transfer, it is com- pletely new. Transmission power flow has to be reversed so that driving torque can be applied to the front wheels. This is accomplished by dividing the standard Turbo Hydra-Matic into two units. The variable stator torque con- verter is attached conventionally to the rear of the engine. However, the gear sets and controlling ele- ments are rotated 180 degreees, are enclosed alongside the con- verter and extend forward along the left side of the engine. A 2- inch wide, multiple link chain ro- Special agent plots overthrow of hidden enemy. The hidden enemy is vapor in automobile fuel lines. engines of the future. Maybe someday he'll help us Causes vapor-lock that stalls cars on warm days. formulate a new kind of fuel for a yet-unknown engine. Our special agent is Dr. John O. Becker, University of How about you? Looking for a challenge—and a Illinois, '64. Here he plots a temperature-pressure-fuel chance to contribute to the exciting new technologies relationship as he specializes in fuel volatility at our shaping tomorrow's world? Your opportunity may be Whiting, Ind., Research & Development lab. One of his here at American Oil. Whether you're a mechanical theories has already been proven. The next step—a engineer, as Dr. Becker is, or a chemist, metallurgist, practical application useful in re-blending gasoline. To mathematician or physicist. make it less prone to vapor-lock. For more information, write J. H. Strange, American In his spare time, Dr. Becker is boning-up on car Oil Company, P.O. Box 431, Whiting, Indiana. A rugged stabilizer bar across the front between the lower control arms, keeps Toronado's front end flat and level when cornering. Integrally d e s i g n e d steering knuckles and plane arms are sim- plified one-piece forgings. Standard power steering for the Toronado is responsive and pre- cise with an overall ratio of 17.8 to 1. Lock-to-lock is about 31/2 turns, compared to nearly 5 turns for conventional steering. Proper steering column angle is achieved with a flexible, constant speed universal joint that also isolates vibrations. A s p e c i a l shock absorber mounted from the frame cross- member at the front to the steer- ing linkage intermediate rod, adds to the car's outstanding steering control characteristics. Supporting front suspension and power train components is a per- imeter "stub" frame terminating with a rear cross bar that supports forward eyes for the rear springs. Box section construction is used for better beaming and torsional rigidity. A new brake drum has been de- spring-loaded downshift detent to Rear suspension consists of a signed for the Toronado with case retain the 'kick-in' feel whenpass- stamped LJ-channel axle with'dead' fins providing quick heat dissipa- spindles bolted on for the rear ing- tion for improved brake life and Another new feature in the lor- wheels. The axle is cushioned by reduced fade characteristics. En- two single leaf springs and four onado is a quiet, draft-free ven- larged brake linings (11x2-3/4" tilation system that eliminates shock absorbers. The leaf springs front and 11x2" rear) are used are anchored with rubber bushings corner-vent windows and exhausts with self-adjusting primary and air under the rear seat and up and are fastened with shackles to secondary shoes. an integral underbody frame at the through louvered outlets below the Impressive looking 15" wheel back w i n d o w . This assures a rear. spiders have functional openings q u i e t e r passenger compartment Two horizontal and two vertical offset into the air stream, so that and reduces wind noise. When the shock absorbers are u s e d to the air flows through the openings side window is partially opened, dampen rear spring wind-up, for and over the cast brake drum fins. air is drawn from the car in the maximum controlled braking, for Tires are 8.85x15. same manner as with a corner- the elimination of wheel hop, and An interesting mechanical inno- vent, with less noise. generally, for smoother, quieter vation for the Toronado is a new Air inlets in the Toronado's ride. Oldsmobile's Toronado is the self-equalizing accelerator cable cowl sides and instrument panel only automobile to use four rear that operates the carburetor. The provide more uniform air distri- shock absorbers. simple throttle device includes a bution. The upper inlets in the instrument panel are blower-fed, enabling comfortable ventilation at low car speeds when ram air is less effective. Toronado also has a retractable head lamp system which is com- p l e t e l y automatic, o p e r a t i n g entirely from engine vacuum. When the head lamp switch is pulled on, the lamps automatically raise to the open position in seconds. Pushing the switch to the park or off position automatically retracts the lamps. A large vacuum reserve tank (600 cu. in.) insures three full cycles with the engine off. P a r k i n g lamps operate inde- pendently. John Lauritzen wanted further knowledge He's finding it at Western Electric When the University of Nevada awarded John Lauritzen ment for the Bell System's revolutionary electronic his B.S.E.E. in 1961, it was only the first big step in the telephone switching system. learning program he envisions for himself. This led him If you set high standards for yourself, educationally to Western Electric. For WE agrees that ever-increasing and professionally, let's talk. Western Electric's vast knowledge is essential to the development of its engi- communications job as manufacturing unit of the Bell neers—and is helping John in furthering his education. System provides many opportunities for fast-moving John attended one of Western Electric's three Grad- careers for electrical, mechanical and industrial engi- uate Engineering Training Centers and graduated with neers, as well as for physical science, liberal arts and honors. Now, through the Company-paid Tuition Refund business majors. Get your copy of the Western Electric Plan, John is working toward his Master's in Industrial Career Opportunities booklet from your Placement Management at Brooklyn Polytechnic Institute. He is Officer. And be sure to arrange for an interview when currently a planning engineer developing test equip- the Bell System recruiting team visits your campus. The Company's first engine, the Wasp, took to the air on May 5, 1926. Within a year the Wasp set its first world record and went on to smash existing records and set standards for both land and seaplanes for years to ' come, carrying airframes and pilots higher, farther, and faster than they had ever gone before. Take a look at the above chart; then a good long look ship in fields such as gas turbines, liquid hydrogen at Pratt & Whitney Aircraft—where technical careers technology and fuel cells. offer exciting growth, continuing challenge, and lasting Should you join us, you'll be assigned early responsi- stability—where engineers and scientists are recog- bility. You'll find the spread of Pratt & Whitney Aircraft's nized as the major reason for the Company's con- programs requires virtually every technical talent. You'll tinued success. find opportunities for professional growth further en- Engineers and scientists at Pratt & Whitney Aircraft hanced by our Corporation-financed Graduate Educa- are today exploring the ever-broadening avenues of tion Program. Your degree can be a BS, MS or PhD in: energy conversion for every environment... all opening MECHANICAL • AERONAUTICAL • ELECTRICAL • CHEMICAL up new avenues of exploration in every field of aero- ENGINEERING • PHYSICS • CHEMISTRY • METALLURGY space, marine and industrial power application. The • CERAMICS • MATHEMATICS • ENGINEERING SCIENCE OR technical staff working on these programs, backed by APPLIED MECHANICS. Management's determination to provide the best and For further information concerning a career with Pratt In recent years, planes powered by Pratt most advanced facilities and scientific apparatus, has & Whitney Aircraft, consult your college placement & Whitney Aircraft have gone on to set already given the Company a firm foothold in the cur- officer—or write Mr. William L. Stoner, Engineering new standards of performance in much the rent land, sea, air and space programs so vital to our same way as the Wasp had done in the Department, Pratt & Whitney Aircraft, East Hartford, 1920's. The 727 and DC-9 are indicative of country's future. The list of achievements amassed Connecticut 06108. the new family of short-to-medium range by our technical staff is a veritable list of firsts in the jetliners which are powered by the highly development of compact power plants; dating back to successful JT8D turbofan. Examples of the first Wasp engine which lifted the United States current military utilizations are the J58- SPECIALISTS IN POWER . . . POWER FOR PROPULSION—POWER to a position of world leadership in aviation. These FOR AUXILIARY SYSTEMS. CURRENT UTILIZATIONS INCLUDE powered Mach 3 YF-12A which recently established four world aviation records and engineering and scientific achievements have enabled AIRCRAFT, MISSILES, SPACE VEHICLES, MARINE AND INDUS- the advanced TF30-powered F i l l variable- the Company to obtain its current position of leader- TRIAL APPLICATIONS. geometry fighter aircraft. IF LOCATION is important to you in choosing your first job, why not talk to the company that has 130 plants and research centers throughout the U.S.A., as well as scores of sales offices from coast to coast? Your placement office can tell you when our interviewer will be on campus. A N EQUAL OPPORTUNITY EMPLOYER DIVISIONS: BARRETT • FIBERS • GENERAL CHEMICAL • INTERNATIONAL • NATIONAL ANILINE • NITROGEN • PLASTICS • SEMET-SOLVAY • SOLVAY PROCESS • UNION TEXAS PETROLEUM There is hardly anything more commonplace Persons skilled in the use of desk calculators in our way of life than the infinite succession of use the reverse of this discovery for purposes of integers, 1, 2, 3, ..., 11, 12, 13,..., 21, 22, 23,.... extracting square roots. Given the problem of de- Yet, our Western civilization did not have this termining the square root of 169, they put the arithmetical notation in general use until about the number onto the lower register of the machine and time of Columbus. Thus, our most intelligent then begin to subtract the succession of odd num- forefathers of seven life spans ago often labored bers. The upper register, of course, keeps track tediously and all but vainly in solving problems of the number of operations that have been per- easily mastered by a sixth-grade student of today. formed. The results yielded by successive steps are indicated in Table II-A, below: We all understand how the integers follow end- lessly, each a single unit greater in absolute value than its predecessor. We often do not un- derstand, however, how easily the natural order of t h e simple i n t e g e r s may be made to do m a r v e l o u s things for us if we handle them properly. For instance, by leaving out all even- number integers, we may establish the equally infinite sequence, 1, 3, 5, 7,..., 2n-l,... . Then by making a summation of the terms in the series, TABLE I below, we arrive at the squares of the integers. Along the first line of the table appear the first ten integers. The second line represents the first ten odd numbers. The third line values are found by performing a running addition of the odd numbers from the line above. These terms, uniquely enough, also represent the squares of the corresponding integers from the first row. The skill of the experienced user of the desk If we restudy TABLE I for a minute, we may calculator does not stop there. Assume the ex- discover the germ of an idea that suggests the ample of determining the square root of 159. He origin of the calculus. Should we inquire, how proceeds just as before, and we pick up his work does the function y = x2 vary when x = 5? By in- in Table II-B on the tenth operation: spection of line three above, we see that y in- creases 9 units between 4 and 5, and increases 11 units between 5 and 6. Or, we may assume that y is changing an average 10 u n i t s per2 unit change of x. Expressed in calculus, y = x , dy/dx = 2x; for x = 5, dy/dx = 10. Further in- spection of TABLE I along line two shows that the "second difference" between the successive squares is simply 2, and it holds for2 the entire range of x. Expressed in calculus, d y/dx 2 - 2, a constant for all x. How may the foregoing be put to work for us? We have merely chanced upon an identity, a mathematical tool which holds for the general case and, therefore, may be applied to any par- ticular case. Probably the most familiar identity In this example, the lower register is heard in the study of mathematics is the example from to "roll over" a long series of nines inasmuch trigonometry. as the subtrahend is larger than the minuend. The desk calculator merely does what it can to (sin2x+cos2x)=l. show a negative number on the lower register. There are a great many more indentities that are equally well known and useful. The undaunted operator recognizes his mis- take by the sound of the roll of the lower regis- The one suggested in TABLE I may be estab- ter and immediately corrects his error by put- lished in may ways. The simplest method likely ting 25 units back into the machine in a process being of addition rather than subtraction. This r e - (x 4 I) 2 = x2 + 2x = 1 stores the line given in Table II-B after 12 Thus we may proceed from 102 to I I 2 by the operations. He then moves the carriage of the simple expedient of adding (2x + 1), or 21, to machine one decimal place to the left and be- 100, the square of 10. We recognize the square gins the sutractive process anew as in Table II-C, below. The shift of the carriage has the of 11 at once as being 121. effect of introducing a factor of ten between the If we rewrite the identity used previously to lower register residues and the consecutive read, odd numbers being subtracted. The difference in (x v 1)2 = x2 + x + (x + 1) place is picked up automatically by the upper and we may use the new form in more difficult as- lower registers. signments. Firs we note that the squares of consecutive integers differ by the sum of the two By this time we are too well educated in the integers under consideration (x + 1)2 - x 2 = process to attempt to subtract 3.5, the next x + (x + 1). The following is a demonstration subtrahend, from a minuend of 0.5 as repre- of the prowess of this variation of the original sented by the balance remaining on the lower identify. register. And as you may have guessed, 12.5 912 = 902 + 90 + 91 = 8,100 + 181 = 8,281. approximates the square root of 159 to three significant figures. We shall have occasion at Similarly, we may use another identity, some later time to look at another method for (x - 1)2 = X2 - x - (x - 1). approximating the square roots of numbers. Solving for 99 2 with the new relationship, 992 = ioo2 - 100 - 99 = 10,000 - 199 = 9,801. There are endless variations: (x + 2)2 = X2 + 4x + 4 1022 = (100 + 2)2 = 10,000 + 400 + 4 - 10,404. And, (x - 2)2 = x2 - 4x + 4 98 2 = (100 - 2)2 = 10,000 - 400 + 4 = 9,604. Repeated application of examples involving use of the simpler relationships spelled out above will help you develop new mathematical skills. Another very useful identity deserves mention- ing at this time. (x + 1/2)2 = x 22 + x 4- 1/4 - x(x t 1) + 1/4 Thus, 1.5 = (l)(2) + 1/4 = 2.25 And, 7.5 2 = (7)(8) + 1/4 - 56.25 CONTINUED TO PAGE 3° Engineering and Science at RCA Neural Networks for speech analysis, are just those required for is used which requires only S transistors. If a solving pattern recognition problems in general. chain of binary stages were used. 22 transistors For a long time machines that recognize speech It is not surprising then, that neural networks would be required. Also, a modified form of have stimulated the imagination of scientists, also show exciting promise in the Reids of this circuit is used lo relate I he horizontal from the engineer to the linguist, both because visual and other kinds of pattern recognition, •canning frequency to the color TV nubcarrier of their potential usefulness to communication as well as speech. frequency. technology and for the formidable technical challenge they represent. Several years of re- Reference—A. L. Nelson, M. B. HtrscHer, T. I). Re/ereme A. J. Hunks anil F. L Johnson, "Novel search at RCA have resulted in notable suc- Manin, H. J. Zadell, J. W. Fuller, "Acoustic Recog- Frequency Dividers lor I (' Sync deiwrators," IV65 cesses in this field by using networks of elec- nition by Analog Feature-Abstraction Ttchnlqutt" I 111 Interntilionut Convention Record, Purl 2, tronic neurons (simulated nerve cells) to iden- Proc. of Symposium on Models of Perception of tify phonemes—the smallest practical units Speechand VisualForm, I4NOV,, 1964, Boston, Muss. Transistorized Portable "Victrolas" into which speech sounds can be divided with- out losing their identity. These neural net- A Novel Frequency Divider for Although transistors have previously enjoyed works operate on the several outputs of a widespread use in portable receivers and mili- spectrum-analyzing filter, dynamically examin- TV Sync Generators tary communication equipment, only recently ing the spectrum and making decisions as to An economical, efficient and high-performance have solid-state devices made any significant phoneme identity. frequency divider circuit for use in new RCA penetration into line operated home instru- color TV broadcast equipment has been de- ment equipment. Advancing device technology veloped. The circuit is a monostable multivi- has made transistor circuitry cost competitive brator with a unique ability to adjust its timing with equivalent tube circuitry, while providing period to be proportional to the period of the improved reliability, instant warm-up, lighter input trigger pulses. The circuit uses only two weight and cooler operation. transistors, and it has the ability to divide an In low-cost phonographs using single stage input frequency by a constant for a wide range tube amplifiers, high-output pickups are re- of input frequencies. It is also quite immune quired. Such pickups are quite stiff mechani- to power supply variations and requires no cally, require a high stylus force, and thus precision capacitors. The circuit requires no track marginally. These low-cost amplifiers externally-applied AFC voltage for regulating ordinarily use "transformerless" power sup- the timing period, such as would be required plies with the attendant design problems of in this application with an ordinary mono- minimizing hum and shock hazards. stable divider. RCA Victor's new transistorized portable phonographs use multistage DC-coupled cir- cuits providing ample power gain for use with pickups of higher compliance and smoother frequency response. Record wear and tracking are thereby improved. The higher efficiency of During recent investigations, 18 consonant the output stage and the elimination of the sounds (for example, /m/ as in "mad" and heater-power requirement result in a cooler /h/ as in "hid") and 10 vowel sounds were amplifier—and make possible the use of a identified with 86% to 99% reliability when secondary winding on the phonograph motor uttered by any of 6 speakers. Machine recog- for the power supply. The resulting isolation nition of consonants is, in general, much more eliminates the shock hazard and makes pos- elusive than that of vowels, since the identity sible the application of conventional ground- of consonants is hidden in the transient be- ing techniques. havior of the spectrum to a much greater ex- tent than in its steady-state nature, as is the To minimize costs and improve reliability, case with vowels. Vowel characteristics, how- this amplifier has been designed to be built on ever, usually are more speaker dependent. a printed circuit board. The need for a separate The recognition performance obtained repre- supporting chassis has been eliminated by sents, by a considerable margin, the best re- mounting the printed board under the turn- sults achieved to date by any investigator. The two periods of a cycle of operation, as table on the record changer motorboard, al- shown in the waveforms, are first, "measure," lowing the output chokes and filter capacitor A "neuron," as used in these networks, is a and then a "recover." When the circuit is in to extend through the motorboard. The motor- simple computing element exhibiting the char- the rest or "stable" state, 01 is saturated and board serves both as a heat sink and mounting acteristics of fan-in and fan-out, an input Q2 is turned off. Once triggered by an input for the output transistors. Volume and tone threshold, and a specified analog relation be- pulse, Q2 is placed in a constant current con- controls are mounted on the motorboard, and tween output and input when the input ex- ducting mode which causes C2 to discharge at all inter-connecting cables and wiring are in- ceeds threshold. An array of several hundred an essentially constant rate. This action is ter- tegral with the record changer assembly. neurons used in speech analysis is structured minated by the next succeeding pulse which in layers; the first layer receives 20 parallel leaves the voltage across C2 at a value directly inputs from the spectrum filter, and by inter- Reference—J.A. Tourtellot, RCA technical report. related to the time period between the pulses. connections among its member neurons makes The capacitor voltage is thus a measure of the elementary decisions about the shape of the pulse repetition interval. The second pulse, These are only a few of the recent achievements which are spectrum. The many outputs of the first layer which terminates the measure period, also indicative of the great range of activities in engineering and pass, in turn, to a second and then to succes- causes regenerative circuit action which turns science at RCA. To learn more about the many scientific sive layers, which make ever more sophisti- Q2 off. Succeeding input pulses cause no fur- challenges awaiting bachelor and advanced degree candi- cated judgments both of the instantaneous ther circuit action until C2 charges (through dates in EE. ME, ChE, Physics or Mathematics, write: characteristics of the spectrum and of the R4) to the point where diode CR2 can again College Relations, Radio Corporation of America, Cherry nature of its changes with time. Finally, binary conduct. The first trigger pulse following the Hill, New Jersey. logic networks make decisions as to the most "recover" period causes the cycle to reoccur. likely identity of the phoneme. A constant frequency division ratio is main- In speech processing, neural networks per- tained over a wider input frequency range than form with great simplicity, limited-accuracy was previously possible as a result of the self- operations on a large number of simultaneous adjusting timing feature. A new color sync inputs, and maintain continuously analog generator, which uses this type of circuit in measures of the reliability of each decision by the frequency divider that relates the horizontal virtue of the analog properties of the comput- and vertical scanning frequencies, is proving ing elements. These properties, so well suited to be highly successful. A 525:1 divider chain CONTINUED FROM PAGE 28 A move advantageous method for finding squares Were we to imagine the example immediately of larger number is presented below: preceding as applying to 75 2 , then the answer (2x)2 = 4(x)2 would be just 100 times greater than the value (3x)2 = 9(x)2 obtained for 7.5 2 . Where a number is divisible by two or three, Therefore, 752 = (7)(8)100 • 25 • 5,625 the square of a smaller number may be deter- 9 5 2 = (9)(1O)1OO + 25 ^ 9,025 mined and then multiplied by four or by nine, And 2052 = (20)(21)100 t 25 - 42,025 whichever is applicable, for purposes of obtain- ing the square of the larger number. Multipli- Proceeding in another vein, if we investigate cation by nine is most easily done by multiply- the successive square of integers in the range ing by ten first, and then subtracting the number from 20 to 30 some additional interesting prop- from ten times itself. erties are discovered. We already realize that For example, (93)2 = 9(31)2 - 9(961) the square of 25 is 49 units greater than the = 9610 - 961 - 8649 square of 24, and that the square of 26 is 51 units Applying the previous method, greater than the square of 25. Hence, the square (144)2 = 4(72)2 - 9(48)2 of 26 is 100 units greater than the square of 24. where either 48 or 72 may be squared by an In TABLE III below, these several squares appear earlier device. We may readily visualize, while the strange but beautiful relationship of one 48 2 - (48 - 25)100 + (50 - 48)2 to one another is indicated. - (23)100 • 4 - 2,304 Thus the integers that are equally far removed And, 9(48)2 = 23,040 - 2,304 = 20,736. from 25 have squares that differ by even hundreds. The subject of the identity Furthermore, the number of hundreds difference x 2 = (x - 25)100 + (50 - x) 2 corresponds with the number of units that the inte- could be dropped at this stage, perhaps having gers are removed from 25. Now, assuming all the proved its worth in yielding all squares between squares up to 25 are known, one may find the 25 and 75 for the mere price of knowing the first square of any other integer from 26 through 75 by two dozen squares. This idea was kicked around using an identity which we write as follows: for a number of years in such a limited state of usage until it was pointed out that an identity x2 = (100) (x - 25) t (50 - x)2 knows no limits. Where we had imposed arbi- First, proving the identity trary limits of usage between 25 and 75, the x2«100x - 2500 + 2500 - lOOx + x2 = X2 identity went blithely on turning out squares of numbers for anyone who consulted its orcle TABLE III powers. Take the example of the square of lH. Properties of Squares of Integers The identity states: „ From 20 to 30 (114)2 = (H4 - 25)100 4 (50 - 114)^ n-5 n-4 n-3 n-2 n-1 n n+1 n+2 n+3 n+4 n+5 = 8900 + (64)2 Now no one is expected to know the square of 20 21 22 23 24 25 26 27 28 29 30 64 in an offhand way, but there is nothing to._kggE. 400 + 500 = 900 one from applying the identity one more tiglgL (114)^ = 8900 + (64)^ , 441 + 400 - 841 = 8900 + (64 - 25)100 + (50 - 64)^ 484 + 300 • 784 = 8900 + 3900 + (14)2 529 + 200 = 729 = 12,800 + 196 = 12,996 5764-100=676 625 Interest in the squares of larger numbers opens several new avenues for use. We can ar- range the problem of squaring a number such as Hence, 382 = (100) (38 - 25) + (50 - 38)2 114 in this format: - (100) (13) * (12)2 = 1,300 + 144 =1,444 (114)2 = (100 + 14)(128 - 14) And, 69 2 - (100)(69 - 25) + (50 - 69)2 • (1OO)(128)+(14)(128)-(14)(1OO)-(14)U4> = (100)(4 (44) + (-19)2 = 12,800 + (14)(28) - (14)(14) - 4,400 + 361 = 4,761. • 12,800 + (14)(14)(2) - (14)(14) Notice how the square of negative nineteen is a = 12,800 t (14)(14) = 12,800 + 196 positive number and forces the identity to yield r e - - 12,996 sults for numbers larger than 50. The general rule we have evolved is (100 +a)2 = (100K100 + 2a) + a2 A similar, but less wieldy, identity may be which becomes upon expansion, written with 50 Occupying the "pivot position" 10,000 + 200a + a2 = 10,000 + 200a + a 2 held by 25 in the example above: and thereby proves the identity. x2 = 200(x - 50) + (100 - x) 2 The range of application lies between 50 and 150, Taking another example with this powerful ne* providing the first fifty squares are at hand. rule, The existence of the identity may be established (151)2 <= (100K202) • (51)2 by expanding the right-hand side. x2 = 200x - 10,000 + 10,000 - 200x + x 2 • 20,200 + 2,601 = 22,801 3 2 CONTINUED TO PAGE T H I N K P O W E R Think diesel power to drive bines and aerospace equipment also are important parts of a truck as big as a house. Think power for the fun of it, to our POWER complex. At IH, POWER is a 2-billion-dollar-a- carry just two people and their camping gear. Think gas year plus business, with research and engineering one of turbine power. Think marine power. Think rockets and mis- our biggest budget items. We need engineers! We espe- siles, and farm equipment and earthmovers. Think about a cially need mechanical, industrial, agricultural, metallurgi- career with International Harvester. Our 4,000 engineers cal, general and civil engineers. If you're an engineering and technicians are thinking power for every purpose from graduate who is intrigued by POWER and its unlimited rocket thrust combustion chambers to gas turbine tractors applications, you should find yourself right at home with us. and trucks. We're the world's largest producer of heavy- Interested? Contact your Placement Officer now for a date to see an IH duty trucks, a major producer of farm and construction representative when he visits your campus. Or if interviews are not sched- uled, write directly to the Supervisor of College Relations, International equipment—and we're doing very nicely in steel. Gas tur- Harvester Company, 401 North Michigan Avenue, Chicago, Illinois 60611. CONTINUED FROM PAGE 30 11,400, 1.14 and 0.000114, all, can be treated as Returning to the example of (114)2, we wish to the same number insofar as mental computation evolve still another general method for finding is concerned. But just as in using the slide rule the square. or a table of common logarithms, it is up to the (114)2 = (111)2 + (111 + H2) + (112 + 113) + user to locate his decimal place at the end of his calculation. (113 f 114) Each pair of bracketed numbers increases the In summary, be the master of the method, rather quare of 111 to the next larger integer. Moreover, than letting the method be the master of you. Keep the sum of the three bracketed quantities is pre- it flexible and fit it to each circumstance by cisely three times the middle or average brack- calling upon the most useful application. After a eted term. Knowledge of the binomial expansion time you will catch yourself using these methods to theorem makes it possible for us to write, (111)2 = 12,321 check answer obtained by conventional means, to which we add, 3(225) - 675 A given identity may be useful in one sense one (114)2 = (111)2 + 3(225) time and in the opposite sense the next. The last = 12,321 + 675 = 12,996 appearing examples are specific applications of Notice how additive processes replace multipli- the more general identity, cation. The intent throughout the use of these meth- (a + b)(a - b) - (a 2 - b2) ods is to substitute a simpler, safer, and surer way where we have been using b as a small integer, of getting mathematical results. The secret is in Writing the identity2 in the opposite sense, recognizing an algebraic process that simplifies (a2 - b ) = (a + b)(a - b), the problem in hand. find the annular area between a pair of concentric The question might arise, "Why so much fuss circles 12 and 20 inches in diameter. Letd andD about the square of numbers?" But in defense, equal the diameters of the smaller and larger cir- call on more and more identities, and use the cles, respectively. squares of numbers to produce many other useful A = irD 2 /4 - TTd2/4 - (n/4)(D 2 -d 2 ) products. For example, (x + l)(x - 112 = x2 - 1 = (iT/4)(D t d)(D - d) = (71-/4)(20 + 12)(20 - 12) And, 17 x 19 - 18 - 1 = 323 » Ur/4)(32)(8) = 64 IT sq. in. Or, 1.7 x 19 = ^ " - 1 = 32.3 Or, find the net cross-sectional area of a water pipe with an 8-inch outside diameter and a 7-inch Again, (x + 2)(x - 2) = x2 - 4 inside diameter. And, 17 x 21 = 192 - 4 = 357 A = (iT/4)(D + d)(D - d) = = (IT/4) (8 + 7)(8 - 7) - 151T/4 sq. in. Notice how easily one can mix decimal frac- The above answers can be further simplified, tions with whole numbers. These methods, as with but it is not our purposes to go into such discus- the slide rule for which they propose to substitute, sion at this time. Other parts of LAY THAT SLIDE tend to treat every number in terms of the sig- RULE DOWN will better serve these purposes, nificant figures that are given. Thus, 114, 11.4, What do you really know about Caterpillar? You think of Caterpillar as a yellow machine, the most modern in the world. crawling along a muddy road. That's all? You've got to imagine engineers, too. Think again. Mechanical, chemical, industrial, metallur- Put 49,000 skilled people on that ma- gical, agricultural, electrical, civil . . . every chine. Add 16 manufacturing plants, in both kind of engineer good schools produce. hemispheres, to your image. Caterpillar has an important position for Still not close enough. all of them. Make that muddy road ten thousand You might check into Caterpillar leader- roads. Mountainous. Jungle. Desert. Snow- ship in the world's business community. We covered. Base camp bouncers in Africa. stood 48th on the latest list of "500 largest Nation spanners in South America. Hill- manufacturers," and we're one of the top and-dalers across Canada. Construction site five exporters in the United States. hair-raisers, so far beyond the reach of cities Ask your placement office for informa- that only the Cat representative bothers to tion on Caterpillar. Find out about some of describe them. the many contributions Cat research engi- Not one yellow machine. Thousands. neers have made in many more fields than Not one Cat representative. An army of earthmoving. them. And a world-wide network of parts Got it? There's a scope of operations suppliers. here that makes Caterpillar a good place to You've got to think of design. Manu- put your life. If you agree, get in touch. We facturing. Sales. And research—an average certainly need what you have to offer! of $40 million a year, spent on research. (In 1964 it was $45 million). Big testing ground facilities. Technical centers, among To Continue To Learn And Grow... . . . is a basic management philosophy at Delco Radio employes through the popular Tuition Refund Program. Division, General Motors Corporation. Since its in- College graduates will find exciting and challenging ception in 1936, Delco Radio has continually expanded programs in the development of germanium and silicon and improved its managerial skills, research facilities, devices, ferrites, solid state diffusion, creative packag- and scientific and engineering team. ing of semiconductor products, development of labora- At Delco Radio, the college graduate is encouraged tory equipment, reliability techniques, and applications to maintain and broaden his knowledge and skills and manufacturing engineering. through continued education. Toward this purpose, If your interests and qualifications lie in any of these Delco maintains a Tuition Refund Program. Designed areas, you're invited to write for our brochure detailing to fit the individual, the plan makes it possible for an the opportunities to share in forging the future of eligible employee to be reimbursed for tuition costs of electronics with this outstanding Delco-GM team- spare time courses studied at the university or college Watch for Delco interview dates on your campus, or level. Both Indiana University and Purdue University write to Mr. C. D. Longshore, Dept. 135A, Delco offer educational programs in Kokomo. In-plant gradu- Radio Division, General Motors Corporation, Kokomo, ate training programs are maintained through the off- Indiana. campus facilities of Purdue University and available to An equal opportunity employer Campus research is important to Detroit Edison Detroit Edison supports research activities at many Research projects such as these provide a chal- engineering universities and receives valuable infor- lenge for both professors and students alike. This mation from these joint programs. Here are some of challenge continues after graduation in our re- the projects now under way at University of Michi- search laboratories and engineering departments gan, Iowa State University, Kansas State University because interesting things are happening at Detroit and Rose Polytechnic Institute. Edison. • Power System Stability Analysis and Generator Field If you're graduating this year and want to put Control your ideas and energies to work —write to Mr. • Analysis of Conductor Vibration George Sold, The Detroit Edison Company, Detroit, • Modern Solutions for Power Networks Michigan 48226. And look for the Edison personnel • Mathematical Models for Peak Load Forecasting representative when he visits your campus. • Block Diagram Representation of a Power Generat- ing Unit—a first step in stability analysis • Thermionic Conversion EXTENSION OF THE PYTHAGOREAN THEOREM The study of higher-dimensional space in (n+l)-space. The analogue geometry is one of the important of the line segment bounded by topics studied in higher mathe- two selected points is a section matics. This study is usually c a r - or piece of that arbitrary n-space ried on using vectors and matrices bounded by some definite, known as the fundamental tools, and while boundary. In the Pythagorean the- desired results can be established orem, the length of the line seg- in a rigorous manner through the ment AB is of fundamental im- use of these tools, this approach portance. This length can be found rarely leads to an intuitive ap- using methods learned in high preciation of the subject on the school, but these methods do not undergraduate level. That is, while generalize easily. A method more the study of higher-dimensional suited to our purpose is integra- geometry is carried on through ap- tion over the arbitrary line in 2- propriate extensions of the basic space, the end points fixing the lim- concepts of plane and solid geo- its of integration. This method metry, the use of vectors and ma- easily generalizes into iterated in- trices rarely results in undergrad- tegration over the arbitrary n- uates intuitively understanding the space, with the boundary of the motivations for the e x t e n s i o n s . the projections of AB onto the co- piece fixing the limits of integra- Thus it is difficult for many under- ordinate axes. (Fig. 1.) tion. This necessarily implies that graduates to treat the geometry of the boundary must be everywhere n dimensions, n > 3, with anything Now let us examine this state- integrable. And just as we must like the " f e e l " with which they ment with an eye toward general- find the length AB in the 1-di- approach problems in, say, solid it izing it. Consider the XY-plane; is a 2-space. A generalization mensional case, so in the n-dimen- geometry. is an n-space, or, what is the sional case we find the amount same thing, an (n+l)-space. We of n-space contained in the bounded A careful study of the theorem shall see it is more convenient to piece, which will be called the"hy- which is the main topic of this use pervolume". T h i s hypervolume paper may aid in the understanding spacetheofsecond one term. Then a sub- lower dimension is has the same fundamental impor- of higher-dimensional geometry. an n-space. Now consider tance as does the length in the This theorem was first discovered ordinate axes; they may be the co- thought case of the line. by Grassman; the statement and of as coordinate n-spaces. proof given in this paper was dis- under C a r t e s i a n geometry For the covered independently by the au- Finally, a reasonable general- thor. This new proof relies en- (n-t-l)-space is spanned by (n+1) independent variables and an n- ization of the idea of projecting tirely upon the very familiar con- space is spanned by n independent the line segment AB onto the co- cepts of the Pythagorean theorem variables; a coordinate n-space ordinate axes is projecting the and the axiom of mathematical in- would be generated by setting any bounded piece of n-space onto the duction. one of the (n+1) variables equal (n-1) coordinate n-spaces. Then to zero and allowing the remain- the extension of the Pythagorean Consider the Pythagorean the- ing n variables to range through theorem is the following, which will orem. It may be stated in the fol- all real values. This procedure, be referred to as the N-Pytha- lowing very general form: In the applied to 2-space, leads to co- gorean Theorem. XY-plane we may take any line ordinate lines or axes; in 3-space and select two arbitrary points A to coordinate planes; and in 4- and B on the line, thus cutting space to coordinate 3-spaces. It off a line segment of finite length. can be seen that there must al- We may project this line segment ways be exactly (n+1) such coord- on the coordinate axes; then the inate n-spaces. square of the length of the line segment AB is equal to the sum of the squares of the lengths of The arbitrary line in 2-space generalizes to an arbitrary n- Sun signals might work fine (except maybe at night) Motorola's sophisticated know-how stems from years but you can count on control when you program with of experience in all major areas of modern elec- the aid of a Motorola transponder and flight com- tronics—from outer space, inner space, and down mand subsystem. to earth—to a company's computer as well as your own dashboard. Besides, if you were skimming along millions of miles in outer space, you'd more than likely feel better In fact —how about your shirt pocket? You might with some sophisticated electronics backing you up. just find a Motorola radio there. What's going on up there? Ask us. Our space boosters have launched every successful non- military communications or weather satellite. Our Saturn S-IVB will power 3 Apollo astronauts from earth orbit to moon orbit. DC-8s have flown more than a billion miles. DC-9s will double the places you can go by jet. Which all adds up to the fact that Douglas can just about double your chances for rapid career advancement, because projects coming up are even bigger than those mentioned. Also, univer- sities offering evening courses toward advanced degrees are close by. And Douglas has a fine scholar- ship program. Let's get together. We are an equal opportunity employer. "What happens if I find, in a year or so, that my real talent and interest lie in another area of electronics-and I want to change direction?" At Sylvania Electronic Systems —with its nation-wide com- seminars, and post graduate study plans conducted on an plex of research and development laboratories and manufac- unusually generous scale. turing plants, this is no problem. The in-house project list At all times, your professional development will benefit by a is diversified enough to accommodate most career changes. positive management attitude toward technical growth as And a comprehensive educational assistance program makes well as the SES small group form of organization —a tradi- it possible to acquire background needed in any discipline. tional small company advantage. And you will be able to Technical or administrative assignments are available in choose one of three distinct routes for advancement, all with current SES project areas including: ground electronics equal rewards—technical specialist, technical manager, pro- equipment for Minuteman missile sites • R&D in electronic gram/project manager. warfare field • electronic security systems • ASW systems • The SES mission—to manage government systems pro- special purpose airborne computers • laser systems • space- grams for General Telephone & Electronics, the parent corpo- borne electronic and optical systems... plus worldwide engi- ration—guarantees an abundant variety of challenging and neering support systems. absorbing assignments in state-of-the-art electronics and Always available-whether you wish to change communications. Wherever your intellectual your field of interest or pursue it —are divi- and professional curiosity leads, you'll have sion-wide conferences, inplant courses and more opportunity to satisfy it at SES. L and the width W; let W be p a r - not be dealing with the arbitrary 3 mensional subspace of E n+1 , allell to P. Then C2 will project solid in E whose volume can be i.e. an arbitrary n-dimensional onto the Xy-plane; see Fig. 2. found be triple integration, but Euclidean space. Let C n be a In the projection W is unchanged with a cubical element of volume b o u n d e d piece of E n whose but the new length is L**. whose sides are respectively par- boundary is everywhere inte- Now introduce a new plane, E . allel to the axes P , Q and R. Let 2 grable so that it is meaningful Introduce a coordinate system of the length L be in the direction to talk of a precise hypervol- 2 orthogonal axes P ' and Q'; let P, the width W and thickness T ume found through iterated in- P' be parallel to P, Q' be p a r - in the direction Q and R respec- tegration. Let C n project onto allel to^the Z axis. Project Cz tively. Either of these axes—let Ei ; call the resulting projec- onto E 2 *; the resulting projection us say Q—must be coplanar with n tions Ci n . Let the hypervolume will also have width W, and length the W axis. It does not matter of C n be H n and the hypervol- L*. which axis is chosen, but once a umes of C j n be Hj n . Then we choice is made we must be con- have: The lines which contain L* and sistent. L** are orthogonal, and these n+1 line segments are projections of Now introduce a new 3-space, (Hn)2= ZZ (Hin)2 3 L. Hence the Pythagorean theorem E . Span it with a coordinate i=l holds and we have: (L)2 •= (L*) 2 + system of 3 orthogonal axes P', (L**) 2 . We multiply each of these Q' and R \ Let P ' be parallel to We shall prove this theorem by lengths by the constant W: (LW)2 P and Q' parallel to the W-axis, induction. We have established it = (L*W)2 + (L**) 2 . But these prod- Our cube in E 3 , C 3 , will project for the case n=l; we will also es- ucts are the areas of C 2 or of its onto both E 3 * and XYZ. Both pro- tablish it for n=2 and n=3 so that projections. Hence the theorem is the reader will have an easier established for n«2 in the special jections will have length L. The time following the proof for the case of projection onto only 2 projection onto XYZ will haveside general case. W*T*; the projection onto E 3 will planes. have side w**T**. The planes of Let E 2 be an arbitrary plane in But now let us bring in the XZ W*T* and W**T** are orthogonal E 3 . C 2 is then a plane figure and YZ planes. We left them out so from the validity of the theorem 2 bounded by a closed curve. H2 for of Fig. 2 because they were un- for n=2 it follows that (WT) - C2 is of course its area, which essential; the reader may now draw (W*T*)2+ (W**T**)2. We multiply 2 may be found by double integra- them in or think about them being by the invariant L-. (WTL) 2 - tion. We can greatly simplify the (W*T*L*)2 + (W**T**L) in. The projection in E2* is also a the theorem holds for n=3 for pro- . Thus proof if we consider not C 2 itself rectangle, and this rectangle will jection onto only two 3-spaces, but a basic element of area(dxdy), which is a rectangle. It can be project onto these other coordin- seen that if the theorem is true ate planes. In these projections Consider the projection onto 3 for this rectangle then2 it must be the length L* will be invariant; E *, W**T**L. This cube will pro- true for arbitrary C , provided W will project to form widths W* ject onto the coordinate spaces the boundary be everywhere in- and W". Then the same argument XZW,3 YZW and XYW. Since Q' tegrable. from the Pythagorean theorem ap- in E * is parallel to the W-axis plies: (L*W)2 = (L*W')2 + and since W** is in the direction 2 Consider the plane E 2 . Let us (L*W") . Substituting, then, we of Q' it follows that W** is invar- introduce a coordinate system of have iant under these projections onto 2 orthogonal axes P and Q. Let (LW)2 2 = (L*W') 2 + (L*W") the three other coordinate spaces. P be parallel to the intersection + (L**W) Then let the projections of the of E 2 with the XY-plane and Q so the theorem holds in full gen- side T**L be T T , T " L " and be coplanar with the Z axis. Now erality for the case n=2. T ' " L " \ The planes of these pro- let our C 2 be a rectangle whose jections a r e all orthogonal so we Consider now the case n=3. We sides are parallel to the axes P can draw no figure, for it is be- apply the validity of the theorem and Q. Let the length of C 2 be yond the powers of any artist to for 1n=2 to s e e that (T**L)%5 (T'L ) + ( T " L " ) 2 t (T'"L'") 2 . 2 draw a four-dimensional figure on Multiplying by the invariant W* a plane. Here is where the reader and substituting, we haye_ must really begin to exercise his (WTL)2 = (W*T*L)2 + (W**T'L') imagination in following the proof, which rests heavily on the reader's (W**T"L") 2 + (W**T'"L'") 2 ability to picture in his mind that so the theorem holds in full ge"* which we will describe. It will be erality for the case n=3. very helpful to keep the proof for The proof in the general case n=2 in mind, and to look for an- is now fairly simple. We know the alogies and points of similarity. theorem is true for some case n*fc In 4-space we have four coor- From this fact we must deduce that dinate 3-spaces. If the axes a r e the theorem is true for the case lettered X, W, Z and W, then ^=(k+l). these coordinate spaces a r e the In (ntl)-space we have (n+1' XYZ, XZW, YZW and XYW. Con- :oordinate n-spaces. One of thes sider now the arbitrary 3-space s spanned by the first n axes E 3 . Span E3 with a coordinate ind only by these axes. Consider system of 3 orthogonal axes P , low the arbitrary space E n ; in" Q and R. Orient P so as to be xoduce a coordinate system o» parallel to the intersection of E 3 orthogonal axes P j , P2, P3» • ' " with XYZ. Now, as before, we will CONTINUED ON PAGE * Why criticize the topless bathing Engineer: " I ' m looking for a Department of unintentional sat- suit. It's just an attempt to keep truly beautiful girl." ire: Readers' Digest, a normally abreast of the times. Co-ed: " H e r e I am." prim and proper magazine, has Engineer: "Okay, you can help finally succumbed to public taste. me look." A recent issue featured an article entitled "The Limits of Intimacy," which was aptly placed on page69. M.E. - - "What's the best way to teach a girl to swim?" Then there was the dumb coed Chem. E. - - " T h a t requires who received a postcard and tore technique. First you put your arm it open it see what was inside. Quote from engineer after Nat. around her waist. Then you gently Sci. 181 lecture: "So that's it! I take her left hand and-- always figured that the stork had M.E.--"She's my s i s t e r . " too small a wingspread to carry Chem. E. - - "Oh! Then you just an eight to ten pound load." push her off the dock." THOUGHTS-- There are many in this world of ours who hold that things break about even for all of us. I have Firth-Mortery: " T e r r i b l y s o r r y observed, for example, that we all History question: " F o r what was to hear you buried your wife last get the same amount of ice. The Louis the Fourteenth chiefly r e - week, old boy." rich get it in the summertime and sponsible?" Wathleywood: "Had to . . . dead the poor get it in the winter. Engineer's answer: "Louis the you know." --Bat Masterson Fifteenth." Some people are always grum- bling because roses have thorns: I am thankful that thorns have roses. --anonymous When the honeymooning EE and his wife were settled in their hotel room, the bride turned to the EE An e n g i n e e r i n g professor and asked anxiously: stopped to talk to one of his stu- "Honey, do you still love me, dents. As they were about to part now we're m a r r i e d ? " Engineer to Co-ed: "Shall we the professor said: "By the way, "Of course," replied the ex- walk or take a cab?" when we met, which way was I playboy. "You know I always liked married women " Co-ed: " L e t ' s walk. I'm too tired headed?" "North," said the student. to get into a cab with you." " G o o d , " said t h e professor, "then I've had lunch!" During an obscenity-possession trial in East Lansing last week, tne prosecutors read aloud to the Two engineers were standing on JWy an entire 187-page book, " T h e a corner and a coed walked by. u One of them said, "Her necks you're Overheard at Coral Gables: "If rgy Boys," seized at the store. gonna say no, say it now Next week's book will be "Fanny dirty." The other answered "Her before I spend all my dough." n » « . Starts promptly at two. does?" Industrial News STUDY LAB FOR MOON EXPLORATIONS - - This unique vehicle is a mobile geological lab- oratory which is expected to contribute valuable scientific information for early American manned landings on the moon. Built by the General Mo- tors Defense Research Laboratories for the U. S. Geological Survey and NASA, the 8-ton vehicle is shown doing preliminary testing in volcanic fields of Southern California. The laboratory will be used to test instruments of the type ex- pected to be installed in U. S. space ships bound for the moon. It will also perform other simulated missions in support of future lunar landings. Laboratory technicians who work with minute mixtures in industry's research and development have a new instrument that can control flows of critical liquids to a microscope drop a minute. Produced from a stainless steel extrusion (right) this needle valve accurately controls liquid flows. It is made by Circle Seal Products Com- pany, Inc. The stainless steel extrusions are supplied by Allegheny Ludlum Steel Corporation. Prospective purchasers of accounting ma- chines can look forward to faster and more flex- ible operation as a result of two significant prod- uct improvements announced by Burroughs Corpo- ration. Using a Burroughs machine equipped with the new Automatic Form Alignment Car- riage, the operator simply inserts an accounting form in the carriage between a pair of form guides. The machine grasps the form and, by sensing minute performations along its right side, aligns the form squarely to the next open posting position. Elasped time averages slightly over '/2 second. Industrial News A unique breakthrough in extremely fast wide band transient instrumentation has been devel- oped at Fairchild's Instrumentation Laborato- ries. These highly refined, ultra reliable, in- struments employ two fiber optics cathode-ray tubes. The newly designed, fully compatible film transport mechanisms permits direct re- cording from the face of the tube. The Fair- child type 977 fiber optics recording oscillo- scope represents a true breakthrough not only in reliability, but also ease of use and reli- ability in catching signal transient recording since the advent of the 24,000 volt cathode-ray tube. A hardened stainless steel impact pin bent and broke, but the glass sphere didn't even crack as a result of implosion tests on a Corning Glass Works underwater float/housing at the Benthos Company of North Falmouth, Mass. In a pressure chamber, the pressure powered pin was activated by a special disc rupturing at 8300 pounds per square inch, driving the pin into the glass with a force of more than 3000 pounds. Because the glass was under compression, a condition in which it seldom fails, the sphere was barely mar- red. Circles show thin smears of metal film left on the surface by the high energy of the impacts. Prototype underwater light design, made by Cor- ning Glass Works for a Perry Cubmarine opera- ting off Andros Island, consists of a quartz io- dine lamp protected by a tubular Pyrex brand glass pressure housing. The light has been tes- ted to pressures equivalent to a 22,000 foot ocean depth. Assembly also consists of a glass reflector, sealed lamp base and electrical con- nector. Metal rods protect the glass assembly from impact. Four of the lights, attached to the outer hull, are being used by the Cubmarine. By changing the lamp, lamp socket and connector, the light design can be adapted for other watt- ages and voltages. National Security Agency . . . where special programs encourage rapid professional advancement for the engineer, scientist and mathematician launching his career. mented laboratories and use of advanced of full-time graduate study at full salary. Near- computer and other equipment, some found ly all academic costs are paid by NSA, whose nowhere else in the world. proximity to seven universities offering i Skilled clerical and technical support will wealth of advanced courses and seminars is free you to concentrate on the most challen- an additional asset. ging aspects of your projects, and thus help IN-HOUSE TRAINING. The new NSA em- speed your professional growth. ployee first attends a six-week general orien- Outstanding Colleagues tation program, followed by a period of spe- cialized classroom study designed to broaden You will work alongside people of enor- mously varied backgrounds and intellectual familiarity with an area or areas of heavy NSA interests, over 500 of whom hold advanced concern (e.g., communications theory, crypt- degrees. analysis, computer logic and analysis, solid state physics). Formal study is complemented Researchers at NSA also receive constant by on-the-job training, as you work and learn stimulus from outside the agency. To assist in under the guidance and direction of highly certain program areas, NSA often calls on experienced associates. special boards of consultants—outstanding NSA is a scientific and technological com- scientists and engineers from industry and PROFESSIONAL ASSOCIATIONS, TRAVEL munity unique in the United States, perhaps The agency fosters a climate of recognition in the world. Unique in its mission, its oper- and advancement for its young professionals ation, its requirements . . , unique, too, in the by encouraging participation in professional scope of opportunity it affords a young scien- association affairs, and assisting you to attend tist or engineer beginning his career. national meetings, seminars and conferences A separate agency functioning within the as well as visit other research facilities where Department of Defense, NSA is responsible related work is underway-government, uni- for developing "secure" (i.e., invulnerable) versity and industrial—throughout the United communications systems to transmit and re- States. ceive vital information. Within this area, which encompasses the whole field of cryptology— Liberal Personnel Policies, Attractive the science of codes and ciphers—NSA project Location teams pursue a broad spectrum of investiga- NSA policies relating to vacations, insurance tions taking them well beyond many known and retirement are fair and liberal. You enjoy and accepted boundaries of knowledge. Be- the benefits of Federal employment without ginning with basic research, these investiga- the necessity of Civil Service certification. tions progress through applied research, Located between Washington and Balti- development and design, prototype engineer- more, NSA is also near the Chesapeake Bay, ing, and on into various phases ot applications ocean beaches, ski resorts and other poular engineering. summer and winter recreation areas, not to At NSA you might specialize in any or sev- mention the varied leisure-time attractions eral of these sectors, depending on your of the nation's capital itself. The location talents and special interests: permits your choice of city, suburban or ENGINEERING. Antenna/transmitter/receiver country living. design . . . high speed computers (digital and analog) . . . transistor circuitry . . . advanced radio communications techniques . . . micro- academic centers as well as from other gov- ernment agencies. wave communications . . . audio and video recording devices . . . cryogenic studies and Career Development Opportunities applications . . . integrated circuitry . . . m i - Your professional growth and earning power crominiaturization. expand from the day you join NSA, without PHYSICS. Solid state (basic and applied) having to accumulate years of "experience." . . . electromagnetic propagation . . . upper NSA career development is orderly and swift; atmosphere phenomena . . . superconductivity substantial salary increases follow as you as- and cryogenics (Ph. D. graduates only). sume more and more responsibility. MATHEMATICS. Statistical mathematics . . . A number of NSA career development matrix algebra . . . finite fields . . . probability programs help shorten the time when you . . . combinatorial analysis . . . programming can contribute at your maximum potential. and symbolic logic. These programs include: ADVANCED STUDY. NSA's liberal graduate Unequaled Facilities and Equipment study program affords you the opportunity to In a near-academic atmosphere, NSA scien- pursue part-time study up to eight hours tists and engineers enjoy the most fully-instru- each semester and/or one semester or more to take a break." This left me with about a dozen workers (mostly freshmen boys afraid to go into an HOMECOMING enclosed space like a lobby with girls around). About 9:00 PM Toni came out with a coke for me, talked for a few minutes about the "boss display" we were making, DISPLAY and bopped back toward the good record music coming from the lobby. By F r i d a y my grades had dropped another full grade-point, and all that remained to be done Editor's Note: While in the key- managed to get some sheet metal was the connection of the water punch room updating the IBM card for the water trough and an elec- and electric motor, and the coat- deck of our subscription list, the tric motor to move the Stalwart's ing of the Osprey with plaster and keypunch p u n c h e d out suddenly arm holding the Osprey in and out c h i c k e n feathers. Some of my "GET SOME BLACK HAIR DYE of the trough. Some of Herbie's regular helpers were hooking the AND A FALSE MUSTACHE." Upon minor assistants had gotten the hose connections to the trough, asking the keypunch why, it punched rest of the needed materials, In- others were mixing the plaster out the tale that follows. c l u d i n g six bushel baskets of and preparing to coat the Osprey, It isn't really my fault that the chicken feathers. In Toni's words, and a group of girls were painting dorm got flooded, although I should the chicken feathers would make the title on the front with various have known that Herbie would have the Osprey "absolutelyrealistic." colors of spray paint. Toni was gotten the connections wrong, and standing next to me, elegant as a The first night while we were queen in her Levi's and sweat- I am awfully sorry that Toni got building the frame, we had about drenched and covered with chicken shirt, talking ecstatically about fifty girls and a hundred-fifty guys us having the best display in the feathers. w i l l i n g to work. Unfortunately, My trouble started when Herbie university. It looked as if things there were only tools and room for were going to be fine, and I felt Wilkes was chosen to head the about twenty to actually work, with pretty good. committee in charge of building the result that fifty girls and a Grody Dormitory's entry in the hundred-twenty guys stood around T h e n I s a w Herbie standing All-University Homecoming Dis- having an informal mixer, while around doing nothing for the hun- play Competition. While all dorm- the rest of us built the frame. Still, dred and thirty-first time, and I itories, fraternities, and s o r o r i - I was heartened by the large num- made the mistake of getting mad. I ties put a tremendous amount of ber of people who did turn out, told Herbie very nastily that he time and effort into their displays, and my night was made worth- should go connect the water intake Herbie wanted to make the great- while when Sandy left the group and output in the utility room of est display of all time. To do this of guys clustered around her for the dorm. To my surprise he went Herbie said he needed " s p e c i a l - long enough to come over to the to do it, although he did give me ists" to handle certain phases of display and ask me how the work a dirty look. the job: A social chairman to line was coming. up a sister dorm and get girls By now the only major thing to working (which would get our boys The second night, while we were do was the feathering of the Os- working), an accounting major to using chicken wire to cover the prey, and two guys had begun to handle finances, and someone in frame and mold the figures, we work on its top and sides. Toni mechanical engineering to take had even more help. There were was all keyed up, and she hopped care of the technical " d e t a i l s . " about three hundred guys from into the water trough and began Naturally Herbie chose me to be our dorm, a hundred girls from putting feathers on the bottom. I the unlucky mechanical engineer. our sister dorm, a hundred more stood back and savored the thought Since our team is nicknamed the guys from other dorms on cam- that my work was almost done. Stalwarts" and our opponent's pus, and fifty girls from a nearby Suddenly water started pouring t e a m the " O s p r e y s , " Herbie's high school, all on the lawn and into the trough! Herbie, who had idea for the display was a thirty street in front of the dorm. Again, been goofing off so much that he toot Stalwart dunking a five foot twenty of us worked on the display. didn't know exactly what stage the Osprey in a trough of running water By the third evening, Wednes- display was in, had thought I wanted ( Stalwarts Douse the Grouse" day, we were ready to begin stuff- him to hook up the water and turn was the title Herbie wanted for the ing the chicken wire with crepe it on! I yelled for Herbie to turn display). I knew the job would r e - paper, and I was hoping that with it off, most of the people standing quire an awful amount of work, and fifty or sixty kids actually work- around just yelled, and Toni just would have refused to do it, ex- ing we could finish this part of stood there watching the water cept that I was going with Herbie's the job in a couple of hours, so splash around her shoes. cousin Toni at the time, and I was that I could study for a math hour Herbie, startled by the sudden afraid Toni wouldn't like it if I test that I had to take the next commotion, ran o u t s i d e and turned Herbie down. morning. We were just getting stepped right on the switch con- Tnere were only five nights started, however, w h e n Herbie trolling the electric motor, and Left to build the display by the came out with a megaphone an- turned it on. With a whir the large nouncing, " i n view of the t r e - figure's arm moved downward, time up and a thesister had been l i n e d dormI had, money gathered. pushing the Osprey and Toni down mendous progress already made, AT THE EXPENSE OF falling seventy refreshments are available in the into the trough. Four or five of us pages behind in thermodynamics, lobby for all workers who want finally moved to try to get Toni. EXTENSION OF THEOREM can be used to show at this point A f t e r we h a d rescued Toni, CONTINUED FROM PAGE 40 screaming, wet, and feathery, I that the theorem is true for this stood in front of the display watch- P n . Let Pi be parallel to the in- case, in the special case of pro- ing the water pour through t h e tersection of E of water was flowing down the coordinate axis which spans E *'l, axis among the axes that span them. corridor from the utility room but one must be chosen, and the Once again we apply the fact that to the lobby, choice must be adhered to consist- the theorem is known to hold for SONOWToni's given me backently. my Let us have P2 coplanar tn one lower dimension. Then mul- p i n , H e r b i e hates me, two dorms with this (n+l) axis, and nlet the tiplying by the invariant W** and full of kids think I'm a complete extent of the hypercube C in the substituting, we find that the the- incompetent, the university has direction of P2 be its width W. orem holds in full generality for threatened to remove my IBM the general case of n*(k+l). card from the file of student en- n Now introduce a new n-space, rollment, and I'm hiding from all E *. Span it with a coordinate Then, under the axiom of math- of them in this keypunch machine. s y s t e m of n orthogonal axes P i ' , ematical induction, the proof is Please, if you can get a false P2', P3', ... Pn'. Let P i ' be par- complete. mustache and some black hair allel to Pi and then let us have dye. leave them on keypunch num- P2' parallel with the (n+l)th axis. ber five late some night. A guy can get very lonely with only a The hypercube n C n nwill project card reader to talk to. onto both E * and F-i . Both pro- jections will have length L. Then the fact that the theorem is valid for the case of one less dimension Advertiser Page Advertiser Page ALLIED CHEMICAL 26 KODAK Inside Back Cover AMERICAN OIL 21 MALLEABLE FOUNDERS 46 ASPHALT INSTITUTE 47 MONSANTO 5 37 BETHLEHEM STEEL 12 MOTOROLA 44 CATERPILLAR 33 NATIONAL SECURITY AGENCY CORNING 8 OLIN- 2 DEERE 8. CO 48 PRATT & WHITNEY 24 & 25 DELCO 34 RCA 29 39 DETROIT EDISON 35 SYLVANIA 17 DOUGLAS AIRCRAFT 38 TIMKEN DUPONT 19 UNION CARBIDE 7 FORD 16 U. S. AIR FORCE 15 GENERAL ELECTRIC Back Cover WESTERN ELECTRIC 23 IBM ]] WESTINGHOUSE Inside Front Cover INTERNATIONAL HARVESTER 31 XEROX 1 Are you ready to test your engineering knowledge with practical problems ? Stretch your mind at John Deere. You'll work on practical solutions to the challenge of developing new machines, materials and manufacturing methods. Vigorous growth in agricultural and industrial markets, both in the U.S. and overseas, offers vast opportunities for John Deere engineers to test their ingenuity and develop their ideas. John Deere has a 128-year tradition of growth and leadership. To continue this record, we need college graduates interested in product design, research, materials analysis, industrial engineering, methods, plant engineering, and reliability engineering. You'll find job opportunities match college majors in most areas of mathematics, science and engineering. You'll work for one of America's 75 largest corporations . . . first in farm machinery sales in the U.S. Yet, you'll retain many advantages of a small company. Our long-time policy of decentralization encourages initiative, ability and leadership. As an engineer, you'll receive project responsibility as rapidly as you can earn it. For more information, check your college placement office . . . or write: Director, College and University Relations, Deere & Company, Moline, Illinois. JOHN DEERE An equal opportunity employer. —we'll promote them both You can talk to some of our chemical engineering big- theory of catalysts to knock a nickel off the tankwagon wigs and come away with the impression that a man who price of a monomer deserves reward comparable to that has not yet forgotten everything he learned in freshman of the grimy one who cuts a plant's downtime in half by calculus is an impractical theorist and a shirker. (Your relocating the filters so that the pump motors quit burn- impression would be wrong. He doesn't mean that at all. ing out. Bessel functions were his meat at one time.) CHOICE is what we believe in offering, in additiori to Others of our boss chemical engineers will sound as the expected emoluments. We believe in it for our own though it is no longer decent for an educated professional good. We can offer it because of our vast and ever-grow- to ook inside a reactor personally. (He neglects to tell ing diversification, which is all from within. We offer you how hard it was to give up a grand time as an ap- choice at the outset and choice after you have had a prentice steamfitter to enter college.) chance to try out your ambitions and see how they work. Observe, then, that both of these types have risen to Drop us a line about your ambitions. If your ambitions bigwigdom. It takes all kinds to run an outfit like ours. happen to be in mechanical or electrical engineering, The chap who applies new directions in the solid state drop us a line just the same. We offer choice there, too. Automating a Complete Steel Mill The automation of Bethlehem Steel Corporation's new Burns tributions are quickly recognized. And, these become startin Harbor, Indiana, cold rolled and plate mills is another giant points to new discoveries and opportunities. Write us now— step toward meeting the demands for stepped-up steel or talk with your placement officer—to define your caree production. General Electric is uniquely equipped to supply interest with General Electric. Section 699-14, Schenectad all the bits and pieces of automation, and to call on and N. Y. (An Equal Opportunity Employer) integrate the skills of more than 120 business departments —skills that run the gamut of specialized and systems engi- neering, manufacturing and technical marketing. Whatever the projects at General Electric, and they are legion, a small- company atmosphere is maintained, so that individual con-