OF TWO LEGENDRE'S COEFFICIENTS. 349 



7. Now let m = 2p + 1 and n = 2q, then q may be equal to but not 

 greater than p. 



Hence by reasoning similar to that in the last case it may be as 

 readily proved that the value of the above general term when /x = is 



l)! (2p-r)\ (2r/) 



r being even ; we may put this under the form 



m ! n\ (m rl)] (n + r) 



(- 1 ) --,-1 



Hence 



m+n-I 



P P.?.. _(-!) 2 v ( m -r-i)! (K 



where m is odd and greater than n, and w is even, and r is equal to 

 the even numbers in succession up to n, bearing in mind that (0)!=1. 



The above is of exactly the same form as in the former case, except 

 that the sign is changed, and the general term may be put under 

 the form 



( - \^~ l 1-3-5 ...(OT-T- 2)1. 3 . 5 ... (n + r- 1) 

 2.4.6...(w + r+l)2.4.6...(n-r) ' 



8. If m be large compared with n and even, the greatest term will 

 be that for which rl, which 



./iv^T*-' 1-3.5 ... (m-3) 1.3.5 ... n 



2.4.6 ...(m + 2)~2.4. 6...(w^l)' 



and all the other terms will be small compared with this term. 



If m be large compared with n and odd, the greatest term will be 

 that for which r = 0, which 



- y^'1.3. 5 ... ( m -2)l.3. 5 ...(n-l) 



2.4.6...(wi+l)2. 4.6...W ' 

 and all the other terms will be small compared with this term. 



