162 NOTE ON A DEFINITE MULTIPLE INTEGRAL. 



Let n be odd, then we shall have 



(K -4). ..3.1 (ft -3) ...4.2 2?r 



' ~ 



Similarly 



'in"-^ dP. = , &c. = &c. 



Lastly j* 



J n 



n-1 



27T 2 



Again, if n be even, we get 



n-: 

 7T 2 



-f-OPr -)-' 



And this, multiplied by Wr , or unity, gives, as before, 



n-1 



and thus (2) becomes, making r = 1, 



n _J(a v cos (9, + ... &c.) 



= -rr/m n - 2 ^^ 1 /(acos^ 1 ) (3). 



V 2 J 



This is the general result given at the end of Mr Boole's paper, 

 and which includes the others. 



