g50 D^ WALLACE ON A FUNCTIONAL EQUATION. 



and e = 13^-, and sm<p=-^^^ , 



and cos(/)= - .^-y^ = 



Now Gos(p= y , and (?^= — ^, 



therefore K«^ + =-^ , and f/0=^^-^-^^ . 



Now cos(/)=l-^ + ^2V:4-TT27^K4:5TF + ^"- 



and 1(.' + = 1 +r^ + l.2^V4+ 1.2.3''4.5.6 -"^^- 



11 1 



Put a for y-g ' C!* for -^ 2 3 4 ' ^* ^^'^ ^^ 2 3 4 5 6 ' ^^'' ^^^ ^® 



have dx= i_c^^2 + c7^43c706:f:&c. ■ '''^'" TTcr^Tc7^* + c;7+&r' 



It is a remarkable property of these expressions, that the coefficients 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 first of these ex- 

 pressions (viz. cos 0) is the secant of (p ; and the law of the terms is known to 

 be this : * 



Let «=1, 



R 2.1 _, 



4.3 a 4.3.2.1 

 '^=172-^-1. 2. 3. 4 "=^' 



. 6.5 6.5.4.3^ 6....1 



^= 172- 'y -TTUTsTl ^ -^ iTTTTe " =^1 ' 



8.7^ 8.7.6.5 8.. ..3^ 8....1 ,„„. 



^=T:2-^-T72T3T4'y-^TT7T6 ^-17^78- =1'"'' 



^=50521, ;3 = 2702765, = 199360981, ;= 19391512145, &c. 

 Then .ec^=.l^^^' + ^-^»' + ^^3^^^^g »' + &c. 



We have now <'»^=<'* {l + T^'?'' +t:^3:4*' + T:273^'tX8 *'**"''•) • 



* EULER, Calculus Differentialis, Pars ii. cap. viii. ; also Legendre, Exercices dc Calcul Integral, 

 tome ii. p. 144. 



