Products of Legendre Functions. Ill 



which must be satisfied identically for all values of yu,. In 

 another form, 



S « 2^1 • ^—l 1 M + N— R) 2 — 4MN — 2(r + 1)(M + N — R) | 



= S^^^^i(M + N-R) 2 -4MNf27'(M + N-Ii)|. 



Clearly the values of r are a, u + 2, ..., alternate functions 

 being missing. The series on the right contains a term of 

 lower index than any on the left unless its coefficient is zero. 

 The indicial equation, for an ascending series of P functions 

 in which the order increases by two is therefore 



(M + N-R)»--4MN + 2r(M + N-R) = 0, 



where r = a. This determines the possible values of a. 

 The equation is a quartic, and becomes, in full, 



(m 2 + m+n 2 + n— a?— «)(m 2 -f- m + n 2 + ;i — a 2 + a) 



= 4(?n s + w)(w , + ; 7i), 



which can be reduced to 



« 4 -« 2 (l + ^m 2 + 2n 2 + 2m + 2^) + (m--n) 2 (m + w + l) 2 ==0 J 



whose roots are 



a — m — n, n — m 7 »i+?i + l, — (m + n + 1). 



Whether m or n be the greater, two ascending series 

 beginning with a function P of positive order are available. 

 If we confine ourselves to integral values of m and n, which 

 are the cases of practical value, we select a.=-m + n + l and 

 u = m — n or n — m, whichever be positive. 



For a descending series, in which the order of the P func- 

 tions decreases by two, we choose ff to satisfy, when r = /3, 



( M + N-R) 2 -4MN-2(r + l)(M + N-R) = 



— the extra term being on the other side of the identical 

 relation. 



This is the quartic 



(m 2 +m+n 2 + n-^ 2 -/8)(7n 2 + ?w + n 2 + n-/3 2 -3 i 8-2) 



= 4(m 2 + m)(n 2 + «) 

 3 D2 



