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4. We may then take 



L, + L 2 



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- 

 mutations as before. 

 5. We must take 



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

 mitting permutations between the values of ^ z l l ... * l l lt or between 

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

 se, and subject to the condition that every such sum as r l i + 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 



fdy dz dy dz\*-i/dy dz dy dz\*\(dy dz dy dz\ Vl 

 \dv dw dw dvj \dw du du dwj \du dv dv dwj 



dz dx dz dx\^fdz dx dz dx\^fdz dx dz da?\ V3 

 dv dw dw dvj \dw du du dwj \du dv dv dwj 



fdx dy dx dy\^/"dx dy dx dy\^/"dx dy dx dy\ v 3 

 \dv dw dw dvj \dw du du dwj \du dv dv dwj 



where \ 1 +X 2 +X 3 =L+p 





These equations are to be solved in every possible manner with 

 permutations between the members of the X set, the p 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- differential part. Let us sup- 

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



