INDUSTRIAL MATHEMATICS 275 



welding and in the design of circuit breakers and other protective devices. Re- 

 cently a mathematical theory has been developed which relates the arc phenomena 

 to the heat transfer characteristics of different gases. This theory has given 

 excellent correlation between the known experimental results, and has enabled 

 very useful predictions of performance under new conditions to be made. The 

 theory has been applied in the design of high voltage air circuit breakers, which 

 are of important commercial value, and it is also greatly curtailing the time and 

 expense necessary to develop many other devices in which arc phenomena are 

 of importance." 



A second example, furnished me by Mr. Reid, has to do with the inter- 

 pretation of wind-tunnel data in aerodynamics: 



(/) "Here it is obviously impracticable to perform full-scale tests of such parts 

 as wings or fuselage, much less of entire aircraft, and the extrapolation from the 

 results of wind-tunnel measurements to the full-scale characteristics of airplanes 

 must be based on theoretical considerations." 



Fourth: Mathematics frequently aids in promoting economy either by 

 reducing the amount of experimentation required, or by replacing it entirely. 

 Instances of this kind are met everywhere in industry, not only in research 

 activities, but in perfecting the design of apparatus and in its subsequent 

 manufacture as well. 



Mr. Alger describes in general terms one situation frequently met in 

 research activities as follows: 



"The first type of problem is one in which there are so many different inde- 

 pendent dimensions of a proposed shape to be chosen, or in general so many 

 independent variables, that it is hopeless to find the optimum proportions by 

 experiment. The truth of this can readily be seen when it is realized that the 

 number of test observations to be made increases exponentially with the number 

 of variables. If 10 points are required to establish a performance curve for one 

 variable, 1,000 observations will be required if there are 3 independent variables, 

 and a million if there are 6 variables." 



As an illustration he cites the following problem: 



(g) "An example of this kind of problem is that of designing a T dovetail to 

 hold the salient poles in place on a high speed synchronous generator. A large 

 machine of this type may have 10 or more laminated poles carrying heavy copper 

 field coils, each assembled pole weighing several tons and traveling at a surface 

 peripheral speed of 3 miles a minute. The centrifugal force on each pound of 

 the pole then amounts to approximately 500 pounds. The problem of designing 

 dovetails to hold these poles in place, even at over speed, is, therefore, one of 

 great importance and technical difficulty. For each such dovetail, there are 

 7 different dimensions which may be independently chosen. While empirical 

 methods have enabled satisfactory results to be obtained in some cases, appli- 

 cation of mathematics has recently enabled marked improvements in dovetail 



