DIFFERENTIAL AND INTEGRAL CALCULUS. 31 



therefore 



P m = - aT 1 V(2* - x*) + aP^ + (m - 1) (2aP w _ 1 - PJ, 

 or mP m = - x m ~ l ^(2ax - x*) + (2 w - 1 ) P m _ 1 ; 



aT 2-l f 2 " x m ' l dx 



therefore .. ^ ax = - - a I 



J o V(2oa?-aJ*) m J 



III. P m =Jx n </(a*-x*)dx 



or (m + 2) P m = - x'"- 1 (a 2 - x' + (m - 1) a 2 P IH| , 



whence f a: m V( 2 - ') ^ = ^ ^ a* [ ^ m " 2 V(" - 



IV. P = 





= - ^'- (2^ - *') 4 aP_ + {aaP^ - PJ, 



whence (m + 2) P w = - x m ~ l (2ax - x*f + (2m + 1) aP^^, 



/* a O^ I 1 f2a 



x m */(2ax-x*) dx=- ^ a x m ~ l ^(2ax - x 2 } dx. 

 o m + 2 J Q 



Another such integral may be given, which really includes 

 these four; or, at least, from it they can all be easily 

 deduced. 



P = J s in w xdx - - /sin" 1 " 1 a; -y- (cos a?) dx 



m - 



= - sin" 1 ' 1 cosaj + (m - 1) /sin m - 2 ic(l-sin a ic) 

 = - sin^ic cosa; + (m - 1} (P m _ 2 - PJ ; 

 whence wP Hl = sin 1 "' 1 ^ coscc 4 (m l) P 7U . 2 , 



