OF TWO LEGENDRE'S COEFFICIENTS. 357 



As before, it may be shewn that if 



.,. - , e . , f 



i/(r)= -,- --.-;- ^ ' , the same function as before, 



(m-n)( 



then /( r ) =i/ (r- 1) <t> (r- l) -^(r + 1) <f> (r + l). 



But in this case we have 



m+n-l 



= (-1) ~{t//(-l)<(-l)-'M + l)<H 7l + 1 )}> as before; 



but ii (n + 1 ) vanishes and ill ( 1 ) = 7-^ T\ > 



' (m n) (m + n+ 1) 



., ,. 1.3.5...W 1.3.5 ...(-!) 

 l = 



1.3. 5.. .m 



13. We will now return to the consideration of 



I S m P n dp and S n P m dp, 

 h Jo 



where m and n are either both even or both odd. 

 First suppose m and n to be even. 



Then 



mn 



|\ Pf ? = (-l) 2 "l-3-5...(m-l)1.3.5.. 

 J m n ^" 2m+l 2.4.6 ...(w-2)2. 4.6... 



w 





(m 



