CIKCUir IIII.OKY ASH Ol'lMAI lOS.M. CALCULUS 759 



M\- i'(|uati(>n ('.\\), tlu- t-xplicii t'(|ii.itiiiii for x., ^ivi-ii <i|HT.iti<>n,illy 

 hy ( 1471, is 



.V, = , / sin u)r. /;(,/- r)(/r = / sin ui(/ — r)^ (r)rfr + //(o) sin u)/ (1.>1) 



wluTf \\'(l)=d (ll //(/). Hy a wi'll known Irinononu-tric formula, tliis is 



.V, =sin ijil I cos a)/. /;'(,/!<// — cos ul / sin ij}l.h'(l)dt + li{o)s\n wt. 

 Writing 



r<//= / rf/- A dt 



Jo »'o •'/ 



this becomes 



Xi=s\n uil I cos o)/. /;'(/)(// — cos w/ / sin a)/./j'(/)d/ 



+h{o)sinwt- J smw{t-T)h\T)dT. (155) 



The first tliree terms are simply the steady-state response to the 

 impressed e.m.f. sin oj/: that is, they represent the ultimate steady 

 state value of Xs when the transient oscillations ha\e died away. The 

 last term, which we shall denote by Ts, represents the transient oscilla- 

 tions which are set up when the e.m.f. is applied. Thus 



Ts=-f sin a)(/-T)/»'(r)(/T. (156) 



.Now from (156) 



T*=-- n h'{T).d.cos'w(T-t) 



UlJl 



and integrating by parts 



^• = i J/^'W+^/°°-^ '^ir-t)%hir)dr. (,57) 



Repeating the process of partial integration, we get: 



Repeating the process again 



_ 1 d .,,, 1 d' ,,,^^ If. , ,, rf» ,, ^. 



