INDUSTRIAL MATHEMATICS 271 



Finally, we must remark upon two facts: (1) that approximate solutions 

 of problems, and hence methods of iteration (successive approximation), 

 play a much more conspicuous role in applied mathematics than in the 

 pure science; (2) that the highly convenient assumption that linear ap- 

 proximations to natural laws (such as Hooke's law and Ohm's law) are 

 sufficiently exact for practical purposes is less often true than formerly 

 was the case, so that nonlinear differential equations are of great importance 

 to the modern engineer. 



Types of Service Performed by Mathematics 



Leaving aside the important but rather trite observation that mathe- 

 matics is a language which simplifies the process of thinking and makes it 

 more reliable, and that this is its principal service to industry, we may 

 distinguish certain less inclusive, but perhaps for that reason more illu- 

 minating, categories of usefulness. 



First: It provides a basis for interpreting data in terms of a preconceived 

 theory, thus making it possible to draw deductions from them regarding 

 things which could not be observed conveniently, if at all. 



(o) An illustration is the standard method for locating faults on telephone lines. 

 Mathematical theory shows that a fault will affect the impedance of the line in 

 a way which varies with frequency, and that the distance from the place of meas- 

 urement to the fault can be deduced at once from the frequencies at which the 

 impedance is most conspicuously affected. This is obviously much more con- 

 venient than hunting the fault directly. 



(b) A second illustration is the mapping of geological strata by means of meas- 

 urements made upon the surface of the earth. One method extensively employed 

 uses a large number of seismographs, each of which records the miniature earth- 

 quake shock produced at its location by a charge of dynamite set off at a known 

 place. A theory of reflection and refraction similar to that used in geometrical 

 optics shows that certain observable characteristics of these records are related 

 to the depth and tilt of the underground layers, and hence enables the situation 

 of these layers to be plotted. By this means the location of the highest point 

 of an oil-bearing stratum can be found, and the most favorable position for 

 drilling determined. 



Underground geology is also studied by means of gravity, electrical or magnetic 

 measurements upon the surface. In this case the basic theory is that of the 

 Newtonian potential field, and the interpretation of the data leads into the subject 

 of inverse boundary value problems, which is still insufficiently understood. 

 Enough progress has been made in several geophysical laboratories, however, so 

 that the gravity method is now being widely used, and the electrical methods 

 appear promising for some applications. , 



Second: When data are incompatible with the preconceived theory, 

 a mathematical study frequently aids in perfectmg the theory itself. The 



