DIFFERENTIAL AND INTEGRAL CALCULUS. 63 



a series which is necessarily convergent since x is < 1 for 

 real values of sin" 1 (a?). 



The same differential equation holds for (sin" 1 a?) 2 for 

 all values of n after 1, but in this case 



/(0)=0, /'(0) = 0, /"(0) = 2; 

 therefore /'- (0) = 2 2 .2, /"' (0) = 4 2 .2 ! .2, and so on, 

 while /"'(0)=0, /'(0)=0, &c., 



so that (sufaO^a. ^ + 2.2 2 ^ + 2.2".4 2 ^j -)-..., 

 (sin' 1 a:)* _ a? 2 a: 4 2.4 a; 6 2.4.6 x* 





_ . .. 



= 2~ + 3 I + 5 6" + 3^78" H 



It is a singular circumstance, that if in (B) every figure 

 whatever (except that in sin' 1 a?) be diminished by 1, we 

 get the series (A). 



For the functions sm(m sin" 1 a?), cos(m sin" 1 a;), we get 

 the equation 



and since f(Q) = and /' (0) = m for the first, and /(O) = 1, 

 f (0) = for the second, we shall have 



sin(m sin" 1 ^) 



= mx - m (m* - I 2 ) ^ + m (m 2 - I 2 ) (m 2 - 3 2 ) f- -,. ., 



L- Li 



= 1 - m' + m 2 (m 2 - 2 2 ) ^ - m" (m* - 2 2 ) (m a - 4 2 ) ^ +..., 



both of which are convergent since x is here also < 1. These 

 series are true for all values of m, but since in finding 

 /()> /'()> we take sin'^O) to be 0, we must always 

 suppose sin' 1 (a?) to be an acute angle positive or nega- 

 tive, according to the general rule already laid down 

 as a convenient universal rule. 



For the expansion of e a s *, we have 



