Products of Legendre Functions. Ill 



order are properly interpreted as zero, 



Q,-oPjog T -^--^- TTr P ?i . 1 +3-^ ri r,_3 



1+/A ; 



Qo^ilogj^ 



a*' 



P?» Qo ~~ Q?«P 



1 P , 2m-5p 



1 . m o . 



"2m — 1 f -r, .1 2m . 2m— 5 



m 



2m — 1 



"S -L »i-lT" 77* o q S T * m— 3 ~t~ • • • 



L o 2m— 2 . z?n — 1 



*/o- 



m 

 Again, 



P»iQi — QwiP i 



p f 2m — 1 2m — 5 ^ '] 



and using the recurrence formula 



2r + l .fi? r = r + l . P^ +1 + rP r _ l5 



we^readily find that P m disappears, and the result 

 t> r> r> p _ 2?n — 1 . 2m — 3 P 



is obtained, — a series beginning with P OT _2. Further, by the 

 recurrence formula, 



P w Q 2 -Q OT P 2 =iP43/^Qi-Qo)-iQ4^Pi~Po) 



— ~9 _ (P»iQl"~QmPl) — 2(P»iQo — QmPo) 



_3//,/2 m-l . 2m- 3\ _1 2m -1 

 ~ T V "3m.m-l" Z^ 1 2 * m 



3/o. 



Reduction of this expression, by use of the recurrence 

 formula in the first term, readily shows that the coefficient 

 of P m _2 now vanishes, and that the final result is 



p n n -p 2 .2m -l.2m -3.2m-5 



3 . m .m — l.m — 2 •"* 



