350 ON THE DEFINITE INTEGRAL OF THE PRODUCT 



Now we have approximately when in is very large and even, 



1.3.5 ...(m-3) _ }_ 1 .3. 5...(m-l) = _!_ 1 



2T4~6"...(TO + 2) ~ (m - l)(m + 2) 2 . 4 . 6 . . . m ~ m 2 /7rm\4 ' 



\2/ 

 and when TO is very large and odd, 



1.3.5...(m-2)_ _1 1 1 



2.4.6... ( +l) m + l|7jm3jp ^ 



If n be also large though very small compared with m, we have 



| P n P n du. = 



Jo 



approximately when m is even, and 



when m is odd. 



TV \ - , TT \ * I 



-I TO* la) n ~ 



ill 



9. The equation P n+1 = (2n+ 1) I P n dfj. + P n _ 1 enables us to express 



An 



P n dfj. m by means of Legendre's Coefficients. 



Similarly 



-p 



" 



-- - 



(27i+l) (2n + 3) (2n- 1) (2n + 3) " (2n- 



Integrate again and by the same relation we get 



+ _p 



- "-' 



(2w-l)(2n+l)(2n + 3) 



1 



(2n-3)(2n-l)(2n + l)' 



