OF THE INVERSE TRIGONOMETRICAL FUNCTIONS. 41 



70. Again, suppose y == sin' 1 a;, 

 therefore ' y + Ay = sin' 1 (x + A), 

 therefore Ay = sin" 1 (x + K) sin" 1 a? 



therefore' _ 



** [(* + A) 



A, 



by Trigonometry, 

 (l - fc+ A)'}] . ' 



put (a: + A) \/(l -* a: 2 ) a: \/{l (# + ^) 2 } = for abbreviation, 

 then 



A?/ sin" 1 a sin" 1 a 2 

 -r* = r = - . Y . 

 Aa; h z h 



A 



A 



(g + A) 2 (l -a; 2 ) - a; 2 {1 - 

 A [(a; + A) V(l - 



_ 



(a; + A) V(l ~ *) + as V(l - (a? + h)' 2 } ' 



thus 



5^ i7* 



the limit of T , when A = 0, is 77^ -- ^ 



ri x \J\\- x) 



or -- 



and the limit of 



sin 



is 1, Art. 21 ; therefore 



dx \/(l ~ z?) ' 



71. Differential coefficient of vers^ar. 

 Let y = vers~ 1 x, therefore 



vers y = x, 

 therefore 1 cos y = x, 



,, - dx 

 therefore -j- = sin y, 

 dy 



, dy 1 1 1 



therefore -,- = -r = ^ ^ x = rr- -- 



dx sm y v(l cos y) VI 1 (1 



1 



