INDUSTRIAL MATHEMATICS 277 



designs to be made. Generally speaking, these improvements have permitted an 

 overall strength increase of 20 per cent to be obtained under steady stresses, 

 and much higher gains to be made under fatigue stress conditions; while at the 

 same time the certainty of obtaining the desired results on new designs has been 

 very greatly enhanced." 



A second example was brought to my attention by Mr. L. W. Wallace, 

 Director of the Engineering and Research Division of the Crane Company: 



(A) "A pipe fitting weighing several hundred pounds and intended for high 

 pressure service had a neck of elliptical cross-section. As originally designed, 

 the thickness of the casting was intentionally not uniform, the variations having 

 been introduced empirically to strengthen it where strength was supposed to be 

 most needed. A redesign carried out on the basis of the theory of elasticity 

 showed the distribution of metal to be inefficient and resulted in a new casting 

 in which the weight was reduced by half, while at the same time the bursting 

 strength was doubled. The method used in arriving at this result is an interesting 

 illustration of sensible mathematical idealization. The casting was regarded as 

 an elliptical cylinder under hydrostatic pressure. As the stresses for this idealized 

 structure were already known, the design problem reduced at once to the simple 

 matter of establishing thicknesses sufficient to withstand these stresses." 



Another example from the field of geophysical prospecting is furnished 

 by Mr. Eugene McDermott, President of Geophysical Service, Inc.: 



(i) 'A specific case of mathematical research in instrument design was recently 

 encountered. The instrument in question was intended for the measurement of 

 gravity. After the machine had been completely built it was found to be unex- 

 plainably inaccurate. After weeks of trial and error it was turned over to a 

 mathematician to try to find the trouble. He soon showed by simple trigonom- 

 etry that the axis of the instrument would have to be located on its pivot with 

 an accuracy which is not attainable. He also pointed out a means of avoiding 

 this feature by a relatively simple change in design, and this appears to have 

 remedied the trouble." 



Another illustration from the petroleum industry, but this time con- 

 cerned with the production of oil rather than prospecting for it, comes 

 from Dr. E. C. Williams, Vice President in Charge of Research of the Shell 

 Development Company: 



ij) "The petroleum industry has one important problem not found in other 

 fields; it has to do with oil production from the ground. A mathematical problem 

 arising from this subject is the following: The oil-gas mixture underground flows 

 under pressure through porous media; with a certain spacing of wells, determine 

 the most economical way to recover this mixture. This is sometimes equivalent 

 to asking: 'In what way can the largest fraction of the oil be obtained over a 

 certain period of time?' Simplified problems of this kind have been solved by 

 potential theorv methods, since classical hydrodynamics becomes too involved. 



