OF TWO LEGENDRE'S COEFFICIENTS. 345 



Therefore in this case 



o 

 But if n be odd, since 



d n ~ l 



and the coefficient of p."~ l in (/A 2 1)" is 



+i 

 nl(-l)' 



i\,/nA t 



/\ 2 ; 



the term independent of p. in *$ is 



1 (n-l)S(-lp 

 2*/H-l 





_ .2.4.6... 



Ipl . 3. 5 ... (n-2) 

 2.4.6...(n + l)- ' ' U ^ 



Hence we see that, when n is odd 



and when n is even 



4. Now take the general case, and suppose m>n, 



* This result was given by Lord Rayleigh in Phil. Trans. Vol. OLX. p. 579, Part II. (1870). 

 A. II. 44 



