g50 DR WALLACE ON A FUNCTIONAL EQUATION. 



ji l + sind) . , c" — 1 



and e =. -^ ; — \-' . and sin ©=—57 — , , 



•*"" 1 — sm0 ^ e^' + l 



, 2e' 2 

 and cosrf)= — ^^- =~- __ : 



. d(b , rfrf) 



Now cos(p=--!—, and rfa'= — 2_ 



" •>■ cos <p 



therefore ^ (e' + O = -^ . and (/(/> = ^^^;_^^^_,^ • 



Now °°^'^ = ^-r^ + TT2-3.4- 1.2.a 4..5.6 +^'^- 



and I (a' + O = 1 +0 + i.2!'3.4 + 1.2.3!V5.6 +-^"- 



Put a for y^ , C. for ^72^374 , 0. for ^ ^ 3^4 5 g , &c., and we 



have dx- i_c^^2 + c.^^-Cc(^= + &c. ' ^~ l + C,x' + G,z' + G,x^ + &c. ' 



It is a remarkable property of these expressions, that the coeflBcients of the 

 terms in the denominators, excepting the signs, are identical ; and it is easy to 

 see that the reciprocals of these series will be recurring series which will have the 

 very same property. The reciprocal of the denominator of the fii-st of these ex- 

 pressions (viz. cos (j)) is the secant of (p ; and the law of the terms is known to 

 be this : * 



Let a=l, 



^=172""=^' 



4.3 o 4.3.2.1 

 'y=l-2-^~ 1.2.3.4 "=^' 



^=172-^- 1.2.3.4 ^-^ 177776"=^^' 



8.7 s 8.7.6.5 8....3o 8....1 ^„„^ 



^=T72-^- i.2.3.4 'y^T7777e^-iT7:78-°"^^^'' 



,^=50521, ,=2702765, = 199360981, /= 19391512145, &c. 

 Then sec0=l + ^./>' + y-^0- + ^2 3'y^g ,3 ./>- + &c. 



We have now dz=d<t> l^\ + ^ci>^ + ^^.-^^ cp^ + ^^^A.^.q "^" ^ ^"^ )' 

 d^ = d. {l-o^-^-rA74'^- 1.2.3''4.5.6 '^^ ^ ^M = 



* EULER, Calculus Differentialis, Pars ii.cap. viii. ; also LECiBNDRE, Exercices dc Catcul Integral, 

 tome ii. p. 144. 



