ON THE PRODUCT OP ANY TWO LAPLACE'S COEFFICIENTS ETC. 373 

 Hence in order to solve our problem, we must find how to express 



p Jp p Jm+p p rl m P tl p P 



(*-*. n.l n Ql f 11 M JT U 4 , , 



- m terms of the form and also to express ~ -(1-ffjT 



in terms of the form , m _ p , multiplied by constants. 



CL^L 



We will now try how far a priori considerations will guide us to 



,Jm+p p .1m. p Jp p 



the form of the coefficient of \'fg*- in the value of -^ ~^i 



The highest index of P x will be n + q, when r 0. 



The coefficient of the corresponding term, where p q and r = 0, is 



1 . 3. 5... (2p-l)l . 3. 5...(2n-l) 

 1.3. 5...(2n + 2p-l) 



d p P d m P , d"P d m P 



But in passing from the value of , q p ~* , m " to that of , p 9 , J 1 , the 



(XjU- CiJJL CljJt. Cip< 



coefficient of the term with the highest index of P will be multiplied by 



2q 1 n + q mp 

 q-p 2n + 2q-l ' 



which equals 



--l! 1.3.5 ...2-l 



(q-p)l 1. 3. 5 ... (2q 3) (n + q-m-p-l)\ 1 . 3 . 5 ... (2n + '2q- 1) ' 



The coefficient of the term which has the highest value of the subscribed 

 index, viz. n + q, will be 



(n + q-m-p)l 1 . 3 . 5 ... (2q- 1) (1 . 3 . 5 ... 2n- 1) 

 (n-m)\ (q-p)\ 1.3.5... (2n + 2q-l) 



The general term must reduce to this, when r = 0. 



Also when p = and m = 0, 



the form of the general term must reduce to 



(n + q-r)\ 1.3.5 ... (2r- 1) 1 . 3 . 5... (2q-2r- 1) 1 . 3 . 5 ... (2n-2r-l) 

 r\ (n-r)l (q- r) I 1.3.5 ... (2n + 2q-2r-l) 



2q-2r+l)' 



