512 



Mr. 0. Heaviside. [Feb. 2, 



impressed force acting at one part of a long telegraph circuit, find the 

 effect produced. One way would be to first find the effect due to a 

 simple periodic force ; from this the effect of an impulse follows ; and 

 from the latter the effect due to any/. A second way is by means of 

 the normal functions, either through the conjugate property or by 

 partial fractions. Lastly, we may decompose the operator Y into 

 operators of the form which would exist were the circuit infinitely 

 long, so that the effect of terminal reflections and absorptions does not 

 appear. Say we have 



F = (Y + Y 1 + Y, + ....)/. (16) 



Then F = Y / will represent the initial wave from the source /, 

 whilst the rest will express the succeeding reflected waves from the 

 terminations of the circuit. The operators Y , &c., may be all of the 

 same type, so that it suffices to solve F = Y /, that is, convert it to 

 an ordinary algebraic functional form, to obtain that form of the 

 complete solution which has the greatest physical meaning, inasmuch 

 as it shows in detail the whole march of F in terms of /. So does the 

 solution in terms of normal functions, but not immediately, because 

 the successive waves are expressed in the form of an infinite series of 

 vibrating systems. Their resultant effect cannot be seen at once. 

 We might, indeed, almost say that the form of solution in successive 

 (or simultaneous) waves was the solution, being of the most explicit 

 nature. Should, however, the impressed force be of a distributed 

 nature, of the type suggested by a normal function, for example, 

 then clearly it is the expression in terms of waves that becomes com- 

 plex and unnatural. We also see that, although a direct transforma- 

 tion from one form of solution to another may be wholly impracticable 

 algebraically, yet it may be readily carried out through the function 

 Y as intermediary. 



Treatment of an Irrational Operator. Solutions in Ascending Series. 



13. The above general remarks are necessarily very sketchy. Some 

 of the matters mentioned may be returned to, but the object of the 

 preceding is merely to prepare the mind of the reader for the more 

 transcendental matter to follow. Let us now consider how to treat 

 irrational operators directly, without the assistance of definite integ- 

 rals. The first form that presented itself to me was that exhibited 

 by 



where p is djdi and R, S, K, L are constants. It occurs in the theory 

 *f a submarine cable or other telegraph circuit, and in other problems. 



