Fundamental Theorem of Invariants, 181 



To find the most general differentiant in question, we must 

 take every combination of the elements whose weight is w and 

 order j, of which the number is obviously (w : i,j), and prefix 

 an indeterminate constant to each such combination ; then 

 operating upon this form with O, we shall reduce its weight 

 by unity, and shall obtain as many combinations of this 

 reduced weight (the order j remaining unchanged) as there 

 are units in (iv — 1 : i,j). Each of these combinations will have 

 for its coefficient a linear function of the assumed indeterminate 

 coefficients ; and in order to satisfy the identity I2D = 0, each 

 such linear function must be made equal to zero. There are 

 therefore (w : i,j) quantities connected by (w— 1 : i,j) homoge- 

 neous equations. Supposing the equations to be independent, the 

 number of the indeterminate coefficients left arbitrary is ob- 

 viously the difference between these quantities, viz. A(w : i,j). 

 The difficulty consists in proving this independence — a diffi- 

 culty so great that I think any one attempting to establish the 

 theorem, as it were by direct assault, in this fashion, would find 

 that he had another Plevna on his hands. But a position that 

 cannot be taken by storm or by sap may be turned or starved 

 into surrender; and this is how we shall take our Plevna. 

 Be the equations of condition linearly independent or not, it 

 is obvious that we must have D(w : i,j) equal to or greater 

 than &(w : i,j). I shall show by aid of a construction drawn 

 from the resources of the " Imaginative Reason," and founded 

 on the reciprocal properties that have just been exhibited by 

 the famous and XI, that this latter supposition, of the first 

 member of the equation being greater than the second, is inad- 

 missible and must be rejected. Observe that (0 : i, j), the 

 number of ways of making up with j combinations of 

 0, 1, 2, . . . i, is 1 ; also that D(0 : i, j), the number of arbitrary 

 constants in the most general differentiant in os to the quantic 

 (a, b, c, . . . l[x, y) { of order j and weight is also 1 ; for such 

 differentiant is obviously \a n . 



Thus we have for all values of w, 



and also 



D(w : i,j)= or>(w: i,j) — (iv — l ; i,j), 



.-. ~D(w: i, j) + D(w-1 : i, j) + D(w-2 : i, j) + ... 

 + D(0: i,j)= or >(w: i,j), 



If in the above condition, for any assumed value of w, > is the 

 sign to be employed, then the equation D(w : i, j) = A(w : i,j) 

 cannot be satisfied for all values of w. If, on the other hand, 



