Industrial Research 



275 



completely baffled. The well-trained engineer of 1940 

 takes it in bis stride and ree:ards it as almost common- 

 place. Tbe well-trained engineer of 1990 will cerlainly 

 regard as equally commonplace the llicory of analytic 

 functions, matrices, and tbe cbaractcristic numbers 

 (Eigenwerte) of difl'erential equations, wbicii today are 

 thougbt of as quite advanced. 



Witb tbis as a background, there need be no apology 

 associated with the statement that such simple processes 

 as algebra, trigonometry, and the elements of calculus 

 are the most common and the most productive in 

 modern industrial research. They freciuently lead to 

 results of the greatest practical importance. The 

 single sideband system of carrier transmissioTi, for 

 example, was a mathematical invention. It virtually 

 doubled the number of long-distance calls that could be 

 handled simultaneously over a given line. Yet the 

 only mathematics involved in its development was a 

 single trigonometric equation, the formula for the sine 

 of the sum of two angles. 



Next in order of usefulness come such subjects as 

 linear differential equations (e. g., in studying the 

 reaction of mechanical and electrical systems to applied 

 forces, the strains in elastic bodies, heat flow, stability 

 of electric circuits and of coupled mechanical sj^stems, 

 etc.); the theory of functions of a complex variable 

 (particularly in dealing with potential theory and wave 

 transmission, propagation of radio waves and of currents 

 in wires, gravitational and electric fields as used in 

 prospecting for oil, design of filters and equalizers for 

 communication systems, etc.); Fourier, Bessel, and 

 other orthogonal series (in problems of heat flow, flow 

 of currents in transmission lines, deformation and 

 vibration of gases, liquids and elastic solids, etc.); the 

 theory of determinants (particularly in solving compli- 

 cated linear differential equations, especially in the 

 study of coupled dynamical systems); and the like. 



Less frequently we meet such subjects as integral 

 equations, which has been made the basis of one version 

 of the Heaviside operational calculus and which has 

 also been used in studying the seismic and electric 

 methods of prospecting for oil; matrix algebra, which 

 has been applied to the study of rotating electric ma- 

 chinery, to the vibration of aircraft wings, and in the 

 equivalence problem in electric circuit theory; the 

 calculus of variations, in improving the efficiency of 

 relays; and even such abstract subjects as Boolean 

 algebra, in designing relay circuits; the theory of num- 

 bers, in the design of reduction gears and in developing 

 a systematic method for splicing telephone cables; 

 and analysis situs, in the classification of electric 

 networks. 



Least frequently of all, but by no means never, the 

 industrial mathematician is forced to invent tecbnitiues 



wbicii till' pure mathennitician has overlooked. The 

 method of symmetric coordinates for the study of poly- 

 phase power systems; the Heaviside' calculus for the 

 study of transients in linear dynamical systems; the 

 method of matrix iteration in aerodynamic theory;* 

 nuich of tbe technique used in the design of electric 

 filters and etjualizers — these may stand as illustrative 

 examples. 



The student of modern mathematics will be impressed 

 at once by two aspects of this review: first, by the heavy 

 emphasis on algebra and analysis and the almost com- 

 plete absence of geometry beyond the elementary 

 grade; second, tb<^ complete absence of the specific 

 techniques which play such a large role in modern 

 physics and astrophysics. It is not easy to say just 

 why advanced geometry plays no larger part in indus- 

 trial research; however, the fact remains that it does 

 not.° As regards modern physics, one may perhaps 

 extrapolate from past history and infer that what is now 

 being found useful in interatomic physics will soon be 

 needed in industrial chemistry. In making this extra- 

 polation, however, it is well to bear in mind that the 

 physics in question is for the most part a mental dis- 

 cipline, its connection with the world of reality still 

 ill-defined and incompletely understood. Therefore it 

 may not prove to be as quickly assimilable into tech- 

 nology as have other disciplines whose symbols could 

 be more immediately identified with experience.* 



Finally, we must remark upon two facts: (1) that 

 approximate solutions of problems, and hence methods 

 of iteration (successive approximation), play a much 

 more conspicuous role in applied mathematics than in 

 the pure science; (2) that the highly convenient assump- 

 tion that linear approximations to natural laws (such 

 as Hooke's law and Ohm's law) are sufficiently exact 

 for practical purposes is less often true than formerly 

 was the case, so that nonlinear differential equations 

 are of great importance to the modern engineer. 



' Heaviside was not himself an industrial employee, but the reformulation of his 

 work in terras of inteKral equations and its interpretation in terms of Fourier trans- 

 forms were both carried out in America by industrial mathematicians. 



< This method was developed in the National Physical Laboratory of England, in 

 the course of studies which in America would probably have been undertaken by a 

 Government or industrial laboratory. 



' Mr. Hall 0. Hibbard. of the Lockheed Aircraft Corporation, comments on this 

 remark as follows: "It is possible that the usefulnessof this principle of mathematics 

 has been overlooked to a large extent in certain fields where it might be applied to 

 advantage. In particular, that phase of engineering known as "lofting," which 

 deals with the development of smooth curved surfaces, might ofTer an interesting 

 field for certain types of advanced geometry. Practically all of this work is now done 

 by "cut and try" methods, and the application of mathematics would no doubt save 

 a great deal of time. The same thing is true in the field of stress analysis, where a 

 great deal of time is absorbed in determining the location and direction of certain 

 structural members. It is even possible that the application of vector analysis 

 technique would greatly simplify certain forms of structural analysis, particularly 

 space frameworks. The lack of application of geometry in these fields is probably 

 due to the wide gap that exists between the mathematician and the 'practical' 

 designer and draftsman, .\dvanced geometry might also turn out to be a very useful 

 tool in connection with problems that we tire no-.v encountering in the forming of 

 flat sheet into surfaces with double curvature, an operation that is extensively em- 

 ployed in aircraft manufacture." 



• In this contieetiou. see the quotation from Dr. E. f. Williamson pp. C84-2S.'i. 



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