Industrial Research 



277 



Types of Service Performed by Mathematics 



Leaving aside the important but rather trite obser- 

 vation that mathematics is a hxnguage which simpHfics 

 the process of thinking and makes it more rehable, and 

 that this is its principal service to industry, we may 

 distinguish certain less inclusive, but perhaps for that 

 reason more illuminating, 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. 



(a) 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 measurement 

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

 which the impedance is most conspicuously affected. This is 

 obviously much more convenient than hunting the fault directly. 



(6) A second illustration is the mapping of geological strata 

 by means of measurements made upon the surface of the earth. 

 One method extensively employed uses a large number of seis- 

 mographs, each of which records the miniature earthquake 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 geo- 

 physical laboratories, however, so that the gravity method is 

 now being widely used, and the electiical methods appear 

 promising for some applications. 



Second: When data are incompatible with the pre- 

 conceived theory, a mathematical study frequently aids 

 in perfecting the theory itself. The classical illustration 

 in pure science is the discovery of the planet Neptune. 

 The motion of the planet Uranus was found to be in- 

 consistent 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 behavior of airplanes with "power on" did not check 

 closely enough with stability predictions which had been made 

 without consideration of the effects 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 longi- 

 tudinal stability with "power on" which enable the aerody- 

 namicist more accurately to predict the stability characteristics 

 of a given design. "Powcr-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 par- 

 ticular 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 predictions 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 understanding of the 

 mechanism by which the reduction was affected and its incor- 

 poration into the theory in a workable form required an extensive 

 and difficult mathematical attack. 



Third: It is frequently necessary in practice to extra- 

 polate test data from one set of dimensions to a widely 

 different set, and in such cases some sort of mathejnat- 

 ical background is almost essential. 



An e.xample of this kind of service, concerned with the 

 theory of arcs in various gases, is furnished me by Mr. 

 P. L. Alger, stafl' 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 behavior of such arcs is of great importance, both 

 in welding and in the design of circuit breakers and other pro- 

 tective devices. Recently a mathematical theory has been 

 developed which relates the arc phenomena to the heat transfer 

 characteristics of different gases. This theory has given ex- 

 cellent 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 de- 

 sign 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 interpretation 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 e.xperi- 

 mentation required or by replacing it entirely. In- 

 stances of this kind are met eveiywhere in industry, 

 not only in research activities but in perfecting the 



