OF TWO LEGENDRE'S COEFFICIENTS. 361 



ml 



(r + l)(2n-2r-l + 2m - 2r) + (m - r) (2n + 2m - 2r + 1 - 2r - 2) 



-2r+l)... (2w + 2m-2r+ 1) 

 2n + 2m - 4r - 1 



l)!(m-r)! (2w-2r- 1) (2n-2r+ 1) ... (2n + 2m-2r+ 1) ' 



rtw 



which is the same in form as the expression for I P n dp. m , taking r + 1 

 in place of r, and m + 1 in place of m. 



Hence the law is generally true and may be written 



f 



J 



-.- 7 . 



r\(m r)\ 



1.3.5 ...2n-2r- 



x - 



16. Differentiating this expression m times we get 



P _ 2 / _ ! v _^il_ 1.3.5...(2n-2r-l) ( , 



' r!(H i -r)!l.3.5...(27i + 2m-2r+l) li 



From the equation 



(2 X+ 1)P..^-^, 



we get 



,7m p ,7m+i D Jm + l p 



/O _L 9n /)<! -L 1 \ - <r +m-2r _ -* +m-2r+i J +m-2r-i 



" "' +r " +I 



Substituting in the expression for P n , we get 



7ii! 1.3.5...(2n-2r+l) 



r-1! m-r+1! 1 . 3 . 5 ... 



i / _ i v _ ml 1.3.5...(2n-2r-l) 



^ 



l p Jm+l p 



* m+m-gr+3 _ * n+m-2r 



-r)! 1.3.5. .. (2i + 2m-2r+l) 



ml 1.3.5...(2n-2r-l) 



3) 



d m+1 P 



A. II. 46 



