INDUSTRIAL MATHEMATICS 269 



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 mathemati- 

 cian is forced to invent techniques which the pure mathematician has 

 overlooked. The method of symmetric coordinates for the study of 

 polyphase power systems; the Heaviside^ calculus for the study of transients 

 in linear dynamical systems; the method of matrix iteration in aerodynamic 

 theory;^ much of the technique used in the design of electric filters and 

 equalizers — 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 complete absence of geometry beyond the elementary 

 grade; second, the 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 industrial 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 extrapolation, however, it is well to bear in 

 mind that the physics in question is for the most part a mental discipline, 

 its connection with the world of reality still ill-defined and incompletely 

 understood. Therefore it may not prove to be as quickly assimilable into 

 technology as have other disciplines whose symbols could be more imme- 

 diately identified with experience.® 



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

 in terms of integral equations, and its interpretation in terms of Fourier transforms 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 govern- 

 ment or industrial laboratory. 



^ Mr. Hall C. Hibbard of the Lockheed Aircraft Corporation comments on this remark 

 as follows: "It is possible that the usefulness of 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 devel- 

 opment of smooth curved surfaces, might offer 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 de- 

 termining 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. Advanced geometry might also turn out 

 to be a very useful tool in connection with problems that we are now encountering in 

 the forming of flat sheet into surfaces with double curvature, an operation that is exten- 

 sively employed in aircraft manufacture." 



8 In this connection, see the quotation from Dr. E. C. Williams on pages 30-31. 



