Products of Legendre Functions. 781 



The next quotient, appearing first in — ■JQ2»»I > 3» is 



2m.2m-\-l-2r— 2.2r-l 

 2m.2m + l-2r-3.2r- 2' 



It is suggested that the general formula is as follows : — 



If 



f(s) = 2»i . 2m + 1 — s . s 4- 1 ?= 2m -s.2/»+s-fl, 



then 



_ in P _^4 r + 3.P ar+1 /(2r) f(2r + 2) /(2r-2) 

 ,Sh.^-* /(2/- + 1) /(2r-l) /(2r+3)V(ar-3) 



/(2r + 4) /(2r-4 ) /(2r + 6) 

 7(2r + 5)7(*r-5)"/<2r+'7)""- 



/(2>— 2»+2) /(2r + 2n) 

 7(2r-2» + l)7C2r + 2»Tf-.i)' 



and the reader will be able to prove by induction, following 

 fchs above method, that this is correct. The analysis is in 

 no way more complicated than that involved in deducing 

 the preceding expressions. 



The coefficient of I D 2r+i in this expression is (4r + 3)S, 

 where 



/(2r-2n + 2) /(2r-2n + 4) ...f(2r + 2n) 

 /(2r— 2n+l)/(2r-2n + 3) . ? ./(2r + 2n + l)' 

 and since 



f(s) = {2m-s){2m + s+l) 

 this becomes 



S = Sib 2 , 



where 



2 771 + 2?— -2» + 3 . 2m + 2r-2n+'5 2m + 2r + 2n + l 



1 ~ 2m + 2r - 2/i + 2.2 m + 2r —2n + 4.'... . 2??i + 2?' + 2n + 2 



_ 1 rO». + ft + r + |) F (m — ?i + r + l) 

 ~~2 ' r(m— n + r + f) ' F(m + w+r + 2) 



and 



2m—2r+2n — j . 2m-2r + 2n — 4 2m-2r- 2n 



S* = 2m — 2r + 2n-l .2m-2r + 2n-~ l d. ... .2ra-2r-2?i-l 



_ 1 r(»i + ?i~r) r(m — ?t-r— j) 

 ~~ 2 *r(wi-n-r) ' r(ro + n— r+i)' 



