INDUSTRIAL MATHEMATICS 267 



automatically that industry relies principally upon the lower branches. 

 What it uses much, ceases by the very muchness of its use to be high. 

 The theory of linear differential equations, for example, is a subject by 

 which the average well-trained engineer of 1890 would have been completely 

 baffled. The well-trained engineer of 1940 takes it in his stride and regards 

 it as almost commonplace. The well-trained engineer of 1990 will certainly 

 regard as equally commonplace the theory of analytic functions, matrices 

 and the characteristic numbers (Eigenwerte) of differential equations, 

 which today are thought of as quite advanced. 



With this 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 frequently lead to results of the 

 greatest practical importance. The single sideband system of carrier 

 transmission, for example, was a mathematical invention. It virtually 

 doubled the number of long distance calls that could be handled simul- 

 taneously 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 systems, 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 complicated 

 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 machinery, 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 numbers, in the 



