IN DU STRIA L MA THEM A TICS VS 



classical illustration in pure science is the discovery of the planet Neptune. 

 The motion of the planet Uranus was found to be inconsistent with the 

 predictions of the Newtonian theory of gravitation, if the solar system 

 consisted only of the seven planets then known. Mathematical investiga- 

 tion indicated, however, that if an eighth planet of a certain size was 

 assumed to be moving in a certain orbit, these discrepancies disappeared. 

 Upon turning a telescope to the spot predicted, the new planet was found. 

 An illustration comes from the aircraft industry. I quote it from a 

 report sent me by Mr. C. T. Reid, Director of Education of the Douglas 

 Aircraft Company: 



(c) "The behaviour of airplanes with 'power on' did not check closely enough 

 with stability predictions which had been made without consideration of the 

 efifects of the application of power; therefore, a purely mathematical analysis of 

 the longitudinal motion of an airplane was carried out, involving the solution of 

 three simultaneous linear first-degree differential equations. The results led to 

 the development of equations for dynamic longitudinal stabiHty with 'power on' 

 which enable the aerodynamicist more accurately to predict the stability char- 

 acteristics of a given design. 'Power-on' dynamic longitudinal stability is an 

 important design criterion in aircraft construction." 



(d) Another illustration arises in communication engineering. Theoretical 

 studies had established the fact that vacuum tubes would spontaneously generate 

 noise because of the discrete character of the electrons of which the space current 

 is composed. The theory predicted how loud this noise would be in any particular 

 type of vacuum tube, a most significant result since it established a limit to the 

 weakness of signals which could be amplified by this type of tube. The predic- 

 tions of the theory were supported by experimental data so long as the tubes were 

 operating without appreciable space charge. But it was found that when space 

 charge was present the noise level fell far below the predicted minimum. In this 

 case the missing factor in the theory was immediately obvious, but an under- 

 standing of the mechanism by which the reduction was affected and its incorpo- 

 ration into the theory in a workable form, required an extensive and difiicult 

 mathematical attack. 



Third: It is frequently necessary in practice to extrapolate test data 

 from one set of dimensions to a widely different set, and in such cases 

 some sort of mathematical background is almost essential. 



An example of this kind of service, concerned with the theory of arcs in 

 various gases, is furnished me by Mr. P. L. Alger, Staff Assistant to the 

 Vice President in Charge of Engineering, of the General Electric Company: 



(e) "An example of this kind of problem is that of the theory of arcs in various 

 gases. It has been experimentally known that the duration, stability and voltage 

 characteristics of electric arcs in different gases and under different pressures 

 vary very widely. The behaviour of such arcs is of great importance, both in 



