392 Royal Society. 



\duj \dvj \clwj 

 \du) \dv) \dw) V 



\duj \dvj \dw) 

 Let the latter class of factors be distinguished into two sets, those 

 where l + m + n = \, 



lli.il— Q m=0 n=0\ 



or /=0 m=l «=0 ) 



\ or 1=0 m = n = \J 



which I shall call uni-differential factors, and those in which 

 l+m + n 7 1, which I shall call pluri-differential factors. 



First, then, as to the form of the general term abstracting from 

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

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



(tT (f)'""(~T^ x r^VY-V"Y-Y"- *-#f (f) ""(/)'"" 



\duj \dvj \dw) \duj \dvj \dw) \du) \dv) \dwj 



/ rf \ V d Y<»2/ d V"2 / d \ e ^ ( d \ e ^( d \\ § 



\Vhfd \ ""»/ d V«n / d \ "*** / d \ ' 3 '»3/_rf \** 



t) \Tv) \jto) " X X (du) \dv) Kdw) * 



\duj 



d\ r ^ 1 



\du) \dvj \dwj J M 



subject to the limitations about to be expressed. 



Call </, + % + ... + '>/, =L, 



>/ 2 + %+... + %=L, 



% + %+... + e %=L 3 , 



and form the analogous quantities M,, M„, M 3 ; Nj, N 2 , 5J 3 . Then 

 we must have 



L 1 +Lj + L a + M 1 + M 2 + M 8 -f-N 1 -f-N 8 + N 3 + j p + g + r 

 =f+g + h + e l + e i + e 3 ; 

 and as the sum of any group of indices /, in, n must he not less than 

 2, we have 



f+g + h + e x + e % + e z +p-\-q-\-r, not less than 2ej + 2e 2 -|-2e 3 , 



so that e, + e 2 + e z must not exceed /+<? + h+p + q + r; furthermore, 

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



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

 integer values from f-\-g 4- A to 2/+ 2y + 2A, and find all possible so- 



