Products of Legendre Functions* 775 



This is essentially Adams' theorem, and the same as 

 Series B. Since P w (/-6), P?^) are polynomials of degrees 

 m, n respectively, and the Q functions are infinite series, we 

 may, by equating coefficients of fi m+n , derive at once 



p 2n\2ml(m + n\y 



riu*yr m \j*) - ^ , n , )2 ^ + %% ^ y, 



giving the complete expansion of P TO P„, and thence the 

 value of the integral 



■I 

 P m P»P r ^ 



I 



-l 

 for all integer values of m, n, and r. The results are known. 



The Second Descending Seines. 

 The second series, if ??i>?* + l, has the indicial equation 

 /3 = m—n—> 1 

 and the form defined by 



with /8=m— w— 1. 



Now again, 



a r -2 _ 2r—3 r + m + n + 1 r—m—n—1 r+m—n 



a r 2r -f 1 ' r-fw + n r — m — n — 2 ' r + m — n — 1 



r + n — m 



and the series becomes r + n m 



= P j.1 2m 2n + 2 2m — 2n-l 2m— 2w- 5 



1.3 2m. 2m -2 2n + 2.2?i+4 

 + 2 . 4 ' 2m-l . 2m-3 ' 2^ + 3. 2rc + 5 



' 2m-2n-2 . 2m-2?i-4 " 2m-2n-l r »»-»-5 + --- * 



