Products of Legendre Functions. 783 



Q 2mi (/a) P 2 »+i(/a) 



- _ X V {£* 4- 1 >> r(m + 7i- r+l) TQ-n + r) 

 " 2*<A ^ r(m-w-r) T(w+n + r-+l) 



r(m-n-r-j) r(m + n -f ?■ + -f ) p . , 



• r(m + n-r + |) * r(m-n + r + i) * W ' 



1 S ,' F( m + n-r + l ) T(m-n + r + l) 



~ 2 , Jo C } V(m - n -V + 1) " r(m + n + r + 2) 



r(ro-n-r + j) r(-m + H+r + f) p . , 



• r(m + n-r + |) * r(m-n + r + -|) * *VV" 



TAg Value of an Integral. 

 We can now determine the value of the important integral 



where a, /5, 7 are integers. The integrand is odd or even 

 in fi according as a + /3 + y is even or odd. Thus the 

 integral is zero if oc + /3-\-y is even. 



When « + /3 + y is odd, various cases arise above. Thus 



i 



n jy -p , T(m + n-r) T(m—n + r + l) 



<42m *2» f 2r+I «> - - r(m _ n _ f) r(m + ?l + r + 2) 



r(m-n-r-i) r(??i+n-|-r-8-|) 



since 



«- 1 %j 1 



Again, 



P O P P ////.- P(^ + »-r+ l) r(m-^n + r) 



1 ^2»i r 2«+l r 2r a/A — — -p, r- — . -pT ; ; — -^ 



J_i r(m— n— r) l(m + n + ?'+l) 



1 \m — n — r — J-) r( m + ?i -f r + f ) 

 * r(m + n — r + lt" I/(>-n-f r + £) * 



There are four such cases in all. We find that they can 



