OP DEFLECTIVE FORCES. 133 



-; 1- . . . , "ii being =a:", the initial value of u ; and Aa:* 



=: Awx"-*4- A^. --^^3 — a:«-2 . . . ; consequently (x4-/0"= 



w(w— 1) „,„ w(?^— l)(/i— 2) ,^_, 

 a:" + nx^'-'h + Ap^x^^/^^ + ^ ^ ^^^^ ^a;«-^/t3 + . . ., 



which is the theorem in question, without limitation. 



279. Lemma C. The fluent /sin'^2d^= 

 -— sin**"* z cos Z+/*— — sin ^^"^ ^d^: ; and % 



sin" xdz = ^- y sin"-'zd^. 



The fluxion d (sin ^-^ z cos z) =sin ^"-^ z^ cos z+ cos 

 zd sin^-*z =: — sin^ zdz + cos% (M — 1) sin^-* zdz = — 

 sinM zdz+ (1 - sin^z) (M— 1) sin m-2 zdz= (m— 1) sin ^-2 



M — 1 



zdz — M sin ** zdz ; consequently sin ^ zdz r= sin*'-* 



1 TT 



zdz d(sinM-*z cos z) ; and in the case of z=— , or 



M <w 



a quadrant, the cosine vanishing, the first term of the fluent 

 vanishes. 



Corollary. When m = 2, the particular fluent 

 becomes^ sin^z dzzz^/'dzzz^ z=^ tt ; when M = 3,y^ 



M — 1 



sin'dznf/" sin 2dz=: — | cos z=:0 ; if M=4, — -~ zz |, 



and the fluent is kT'-^'i in the same manner for Mri6, 



we have ' ' .-^ ; and the series may be continued at 

 pleasure for all the even values of M.] 



