344 ON THE DEFINITE INTEGRAL OF THE PRODUCT 



Integrating with respect to x, we get 



d n P x (1 T'-'V' +1 d n+l P x 



but ( 1 - a?)" ^ <& = - / 7-n j +T ' ( see P- 



v ; + 



. 

 dx 2i-lm + 2 da? ' 



'P,' 



" +I 



When /*=!, x=l and all the terms vanish; also when fj.= I, x=l 

 and all the terms vanish. 



3. By means of equation (4) of Section I., viz., 



we may at once prove the theorem 



o 



[i fi 



or P n .P,,_ 1 dp.= ] P m .P n .jdu when n is even. 



Jo Jo 



Write S n for I P n dfi, which vanishes when p. = I . 



Then 



Multiply hy P n = -. " and integrate ; 



p n . p n _^ 



fl 0,j il fl 



P..P,, + ^--2i(flLVt+ 



JO * JO 



If n be even, $ will involve only odd powers of /A and will con- 

 sequently vanish when p. = 0. 



