354 Mk MURPHY'S THIRD MEMOIR ON THE 



Now it is obvious by putting m = 1, 2, &c. successively, that the 

 finite series is always =0, and therefore the infinite series [which is 



( — t'Y 

 the same as the coefficient of ^ ^ — ^ in the expression for ( — 1)°^„] 



vanishes also, so that if V„ be arranged according to the powers of t', 

 it is + 0. ^' +0^'^ + &c,, nevertheless its value is in one instance infinite, 

 namely, when t = 0, for then P„ = P„+i = &c. = 1, and therefore 



F„ = l + (2. + 8) + ("^ + f^f-^^^^^" + ^)^fV")(^^ + ^) + &c. 



= (l+A)(]-A)-«''+^ when h is put =1. 



= X . 



And if V„ did not possess this infinite element ft Vj", from i> = 

 to t = 1 would vanish, whereas its actual value is the same as 



26. To express the transient Junction Vn in a finite form . 

 Since by Art. (24.) K = P, + (2w + 3)P„+, 



, (2w + 2)(2w + 5) (2« + 2)(2w + 3)(2w+7) « „ 

 "• j~^2 ■ '^^ 2 2 3 • "..+3, «c. 



therefore 



1.2.3...(2w+l) r'„ = l .2. 3.. .2?? X (2« + l)P„ 



+ 2.3...(2w + l) X (2w + 3)P„+,./f + 3.4...(2w + 2) x (2« + 5)P,+s^\ &c. 

 when A is put equal to unity. 



But in general, 



{1 - 2A (1 - 20 + K"] -^ = P„ + P,h + P^A^ + ...P„A" + P„+, A"*' + &c. ; 

 c?^''^"{l-2^(l-2j?)+^'|-i^ 



= 1.2.3...2?«.P„ + 2.3,.,(2w4-l).P„+, A + 3.4...(2« + 2).P„+2AH&c. 



