280 hELL SYSTEM TECHNICAL JOURNAL 



for the impossible; for many desirable objectives in industry are as un- 

 attainable as perpetual motion machines, and frequently the only way to 

 recognize the fact is by means of a mathematical argument. 



{o) A certain type of electric wave filter which is usually referred to as an 

 "ideal" filter would be very useful if it could be produced. However, it has been 

 shown mathematically that such a structure would respond to a signal before the 

 signal reached it; in other words, that it would have the gift of prophecy. Since 

 this is absurd, it follows that no such filter can be built, and consequently no -one 

 tries to build it. 



Still another example from the field of communication deals with the 

 design of feedback amplifiers. 



(/>) In practice, any amplifier is intended to handle signals in a given frequency 

 band. For various reasons, it is preferable not to have it amplify disturbances 

 outside this band, and hence its gain characteristic is made to drop off as rapidly 

 as possible outside the limits of the useful band. It has been shown theoretically, 

 however, that the gain cannot decrease at more than a certain rate, which can 

 easily be computed, without causing the amplifier to become unstable. As a 

 matter of fact, the allowable rate at which the gain may fall is often surprisingly 

 low, and a great deal of design effort would be wasted in the attempt to obtain 

 an impossible degree of discrimination if the theoretical limitations were unknown. 



Eighth: Finally, mathematics frequently plays an important part in 

 reducing complicated theoretical results and complicated methods of 

 calculation to readily available working form. So many and so varied 

 are the services falling in this category that it is difiicult to illustrate them 

 by means of examples. We arbitrarily restrict ourselves to two, chosen 

 primarily for the sake of variety. The first comes from Mr. Hibbard: 



((/) "In aircraft design the metal skin, though thin, contributes a large part of 

 the structural strength. Nevertheless, such thin metaUic plates will buckle or 

 wrinkle after a certain critical load is exceeded. Beyond this point the usual 

 structural theories can not be applied directly and it is therefore necessary to 

 introduce new methods of attack to predict the ultimate strength of the structure. 

 These stiffened plates are difficult to deal with theoretically, but by interpreting 

 the effect of the stiffeners as equivalent to an increase in plate thickness or a 

 decrease in plate width, the calculations can be brought within useful bounds." 



The reduction of electric transducers to equivalent T or 11 configurations, 

 the interpretation of the elastic reaction of air upon a microphone as equiva- 

 lent to an increase in the mass of its diaphragm, the postulation of an 

 "image current" as a substitute for the currents induced in a conducting 

 ground by a transmission line above it, and a host of other common pro- 

 cedures could be cited as similar instances of simplification based upon 

 more or less valid mathematical reasoning. 



