50 DIFFERENTIAL AND INTEGRAL CALCULUS. 



Now a #=cos7r- 



if tan </> = - , and p = V(l - x + a? 2 ), 



2 -l 1 sin n 



dx n 2 V(- 1) p sin ' (a? - x + l) n+1 



sin (ft + !)</> [2 

 sin /3 H+i! 



where tan< as above = ' , or since sin< = / , ^ 



Hi 



we may write the result \ii 



\ a -- j 



This method will apply to all such cases of unreal 

 factors. 



The n th differential coefficient of tan" 1 ^ is most con- 

 veniently expressed in terms of JTT tan" 1 ^ which call 0, 

 then x cot 0, 



and -j- = t 5 = sin 2 : 



dx l+x* 



therefore -~ i =2 sin cos -^- = - sin 20 . sin 2 ; 

 dx ctx 



dx 1 



because js = ^T/I ? 



(zi/ sin i/ 



73 J^3 



therefore -A = - - 7 J2 cos 20 sin 2 -f 2 sin 20 sin0 cos 01 

 dx 3 dx l 



= sin' 2 0.2 sin (sin cos20-f cos0 sin 20} 

 = 2 sin 3 sin 30. 

 We observe the law so far to be 



-^ = (- i)"' 1 [n l sin M sin M 0, 



and assuming this to be true for any particular value of ft, 

 we at once see that it is true for n -f 1 ; therefore, &c., the 

 ordinary course of the proof by " Math. Induction.'' 



