221 



the numerical coefficient and the uni- differential factors (except of 

 course so far as they enter into J). This will be as follows : 



/ 



// \ *i f (L N ^m\ X / \ t f d \ i f d \^'"i f (J \*^i f // \ i f fJ \ *t / // \ ^i 



^/ (rf^J irfic/ X VrfwJ V^iy fc/ X " V^M/ V^/ \5w/ ' 



' d \ l *( d \ lTO 2/ (?\ ln a / rf\ '^ / d \ 3m ^/ d\ 2n * 



duj \dvj \dw) \duj \dvj \dwj 



J Y J (-J-J z* x (j") (~) ( ) 3<2r 



x [ - ) [ ) ( }S x , 

 \duj \dvj \dwj J w 



subject to the limitations about to be expressed. 

 Call if + 8/ + _.4_i/ = L 



and form the analogous quantities M lt M 2 , M 3 ; N lt N 2 , N 3 . Then 

 we must have 



and as the sum of any group of indices I, m, n must be not less than 

 2, we have 



f+g + h + e l + e 2 + e s +p + q + r, not less than 2e 1 + 2e 2 + 2e s , 

 so that e 1 + e. i + e 3 must not exceed f+g + h+p + q + r; furthermore, 

 p + q + r must not exceed /+ g + h ; and finally, 



1. We may first take e 1 + e 2 + e 3 =E, giving to E in succession all 

 integer values from/+^-fAto 2/+2y + 2A, and find all possible so- 

 lutions of this equation with permutations between the values of 

 <?!, e 2 , e 3 . 



2. We may then take p-\-q + r=s, giving 5 in succession all in- 

 teger values from 1 tof+g + h, and find all possible solutions of this 

 equation with permutations between/", g, h. 



3. We may then take L + M + N=/+ + A + E s, and find all 

 the values of L, M, N, with permutations allowable between the 

 values of L, M, N. 



