DIFFERENTIAL AND INTEGRAL CALCULUS. 91 



and let (1) # = tan0, (2) y = z cos"0, then 



dy__dy_dQ__d l 1 , ^ _ ^. 



fo d0 6fo~d0'l+a; 2 ' } dx~d6 J 



therefore l + 



c d0* 1 + a? 



.,v 9 eP . du 



1+* 



also , 2nx(l + x*} c -/ = 2n -^ tan (9. 



aa? at/ 



therefore 



or 1+^ -f 2 n + 1 ^ = 



and the equation after the first change is 



C -J 2 cos 2 + 2n sin cos 6 -^ + n (n 4- 1) ?/ = 0. 



Next, = cos 71 6> - w cos"" 1 ^ sin (9, for y = z cos" ^, 



+ nz {(n - 1) cos"- 51 sin 2 - cos" 0} ; 



therefore cos*0 + 2w sin0 cos0-^ 



= cos n+2 2 - 2n cos" 41 sin + ^ cos" (n sin 2 - 1 ) 

 + 2n cos n+1 sin -^ - 2n 2 2 cos" sin 2 : 



also n (n + 1) y n (n + 1) z cos n ; 



therefore, adding, 



= cos"" ^ + n*s cos" 0(1- sin 2 0) = cos" +2 S + n*z I , 



