(/A'i(7/ IIII.OKY .l.\n OI'I.R.I I IHX.II. (•.//.( 7 /.C.V 729 



This is a ver>' important theorem in connfction with transmission 

 hne problems where retarihition, (hie to finite velocity of propagation, 

 occurs. Its priM)f proceeds as follows: 



If the auxiliar\' function k = k{t) is (lit'iiud i>y the operational 

 equation 



k=e >-!■ 



then !>>■ rheorem l\ . 



g{l)=jJ^k(T)h{t-r)dr. (a) 



Now, corresponding to the operational equation k = c'^'' we have 

 the integral equation 



^- = / k{t)e f<dt. 



P ^0 



The solution of this integral equation, which is easily verified by 

 direct substitution in the infinite integral, is 



k{t)=o for /<\ 



= 1 for />X. 



Hence equation (a) becomes 



g{t)=o (or i<\ 



= x / h{t-T)dT for />X 

 dU\ 



= h{l-\) for />X. 



Theorem IV, employed in the preceding proof, as stated above, is 

 extremely important and w^e shall have frequent occasion to employ 

 it in specific problems. We shall now apply it to deduce an important 

 theorem which extends the operational calculus to arbitrary impressed 

 forces, whereas heretofore the operational equation h = \/II{p) applied 

 only to the case of a "unit e.m.f." impressed on the system. 



It will be recalled from a previous chapter that if x{t) denotes the 

 response of a network to an arbitrary force /(/), impressed at time 

 1 = 0, and if h{l) denotes the corresponding response to a "unit e.m.f.," 

 then 



and 



x{t)^^J\{r)S{t-T)dr (31) 



\-=rhit)e->'dt. (30) 



pllip) 



