Royal Society. 393 



lutions of this equation with permutations between the values of 

 e lt e 2 , e 3 . 



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

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

 equation with permutations between f, g, h. 



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

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

 values of L, M, N. 



4. We may then take 



L, +L 2 +L 3 =L 



M 1 + M 2 + M 3 =M 



N 1 +N 2 +N 3 =N, 

 and solve these several equations in every way possible, with per- 



and solve in every possible manner these equations, but without ad- 

 mitting permutations between the values of '/j B J, .. .*'/,, or between 

 the values of the members of the other of the third sets taken each^jer 

 se, and subject to the condition that every such sum as r l i -Y r m i -\- r n i 

 must be greater than unity. Every possible system of values of 

 these nine sets will furnish a corresponding pluri- differential part 

 to the general term. 



Next, as to the uni-differential part, we may form the quantity 



dy dz \"i 



(dy dz dy dz\**/dy dz dy dz\^i(dy dz 



\dv dw div dvj \dw du da dwj \du dv 



(dz dx dz dx\*-*(dz dx dz dx\^(dz dx dz dxV* 



\dv dw dw dv) \dw du du dw) \du dv dv dw) 



(dx dy dx dy\ k t(dx dy dx dy\^(dx dy dx dy\"3 



\dv dw dw dv) \dw du du dw) \du dv dv die) 



where \ l +\ 2 + \ 3 = L+p 



Hi +/j, + f j i = M + q 



><, + .<„+, ' 3 =N 4- r. 



These equations are to be solved in every possible manner with 

 permutations between the members of the \ set, the fj. set, and the 

 v set. Finally, we have to consider the numerical coefficient. To 

 give a perfect representation of this, we must ascertain what identi- 

 ties exist in the factors of the pluri-differcntial part. Let us sup- 

 pose that one set of operators upon x is repeated 0, times, another 



W, times, and so on, giving rise to the powers 0,, 2 8 a in the X 



line. Similarly, form <£,, ifo, ...typ from the y line, and \p t , \p. 2 . . . \p y 



