INVERSE METHOD OF DEFINITE INTEGRALS. 369 



eos{n+2)(l)+'^r— .h cosn(f)+- -^ .k^cosin-2)<p+ . . . —— .h" cos{n—2)(p+ h"*' cosn<p 



^ fTT2rcos2^1~Fp^' ' 



Hence, 



costKj) .{I -h"*^)+h\——cos{n-2)(p-cos{n+2)<p\+hH— — ~-cos(w-4)0 — — cosw0> 

 ^" {l + 2Acos20 + ^^}»+' ' ^' 



when h is put = 1. 



Thus F = (^-^)(^ + ^^ 

 which is evidently a transient function, as its general value for A = 1 



7r 

 2 



is zero, except ^ is an odd multiple of -, when its value becomes 



infinite. 



And in general F„' and F„" are equal, when h is put equal to 

 unity, and therefore F„ has a factor 1 — A in its numerator, which causes 



TT 



its general vanishing state, except when ^ = „, or an odd multiple of 



^, when the denominator becomes (1-A)^"'*"^ and as the numerator is 



of only n + 2 dimensions, it is evident F„ in this case is infinite, when 

 k= I. 



In general f , ~i . r.^ 1: 2 = 2 tan"' . I^^^ . tan 0> + const., 



which taken from = to ^ = - is equal to tt, a quantity independent 



of h, a result similar to those already obtained from other transient 

 functions. 



39. When the sum of a series containing transient functions is 

 required, the following process, with only such modifications as may 

 simplify particular cases, will apply. 



