INDUSTRIAL MATHEMATICS 279 



scribers fluctuate from moment to moment, the number of switches required de- 

 pends in part upon the height to which the crests occasionally rise in this fluctu- 

 ating load. It is also influenced, however, by the way the trunks are arranged, 

 by the order in which the switches choose them, and by many other factors. 

 Experimental appraisal of the effect of these various factors is impossible, both 

 because it would be very costly, and because it would be exceedingly slow. Mathe- 

 matically, however, they have been studied by the theory of o priori probability,^ 

 which is used not only in determining how much apparatus to install in a working 

 exchange, but also in comparing the relative merits of alternative arrangements 

 while in the development stage. 



Sixth: Mathematics is frequently useful in devising so-called crucial 

 experiments to distinguish once for all between rival theories. A famous 

 example in the field of physics was the study of the refraction of starlight 

 near the sun's disc, which afforded a means of deciding between Newtonian 

 and relativistic mechanics. In this case, mathematical investigation 

 showed that the result to be expected was different according to the two 

 theories, and astronomical observations confirmed the prediction of rela- 

 tivistic mechanics. In the industrial field, an example of this kind comes 

 to me from Dr. Joseph A. Sharpe, Chief Physicist in the Geophysical 

 Laboratory of the Stanolind Oil and Gas Company: 



(«) "As an example of the second sort of use of analysis there is the case of 

 our study of 'ground-roll,' the large amplitude, low frequency surficial wave 

 which caused so much grief in the early days of seismic reflection prospecting 

 when filters were not used as extensively as at present. We hope to use our study 

 of this wave motion as an aid to a better understanding of the properties of the 

 surficial layers of soil and their effects on the reflected waves in which we are 

 primarily interested. 



Two views on the ground-roll are current, although neither is based on very 

 much observation, and this of an uncontrolled sort. One view states that the 

 ground-roll is an elastic wave. Analysis predicts that this wave will have a 

 certain velocity in relation to the velocities of other waves, that it will have a 

 certain direction of particle motion and relation of maximum horizontal to maxi- 

 mum vertical component of displacement, that it will attenuate with distance 

 according to a certain law, that it will attenuate with depth in a certain way, 

 and that its velocity will follow a certain dispersion law. The second view main- 

 tains that the 'ground-roll' is a wave in a viscous fluid, and analysis predicts a 

 behavior which is similar in certain cases, and different in others, to that of the 

 elastic wave. Having the predictions of the analysis at hand, we are enabled to 

 devise a group of observations, and the special equipment for their prosecution, 

 which will provide crucial tests of the two hypotheses." 



Seventh: Mathematics also frequently performs a negative service, but 



one which is sometimes of very great importance, in forestalling the search 



^ Not statistics, which is a posteriori probability. This is one of the few cases in industry 

 where the a priori theory finds application. 



