Proof of the Fundamental Theorem of Invariants, 179 



and thereby to advance the standards of the Science of Alge- 

 braical Forms to the most advanced point that has hitherto 

 been reached. The stone that was rejected by the builders 

 has become the chief corner-stone of the building. 



I shall for greater clearness begin with the case of a single 

 binary quantic {a, b, c, . . . , llj^x, y)\ Any rational integral 

 function of the elements a, b, c,. . . I which remains unchanged 

 in value when for them are substituted the elements of the 

 new quantic obtained by putting x + hy instead of x in the 

 original one, I call a Differentiant in x to the given quantic. 



By a differentiant of a given weight w and order j, I mean 

 one in every term of which the combination of the elements 

 is of the jth. order and the sum of their weights w, the weights 

 of the successive elements (a, b, c,. . .1) themselves being 

 reckoned as 0, 1, 2, . . , i respectively. 



The proposition to be proved is, that the number of arbitrary 

 constants in the most general expression for such differentiant 

 is the difference between the number of ways in which w can 

 be made up with the integers 0, 1, 2, 3, . . . i (repetitions 

 allowable), less the number of ways in which w — 1 can be 

 made up with the same integers. We may denote these two 

 numbers by (w : i,j), {iu — 1 : i,j) respectively, and their dif- 

 ference by A(i«? : i,j). Then, if we call the number of arbi- 

 trary constants in the differentiant of weight w and order j 

 belonging to a binary quantic of the ith order D(zy : i,j), the 

 proposition to be established is that D(iv : i,j) = A(w : i,j). 



Let us use £2 to denote the operator 



d rt7 cl ., d 



a^T+2b-y + . . . + ik -=-,, 



do dc at 



and to denote the operator 



Then it is well known that the necessary and sufficient condi- 

 tion for D being a differentiant in ^is that the identity OD = 

 be satisfied. 



Let us study the relations of H and in respect to D. 



In the first place, let U be any rational integral function of 

 the elements of order j and weight w ; then I say that 

 ft.O.U-0.n.U = (ij-2iv)TJ. 



For if we use * to signify the act of pure differential opera- 

 tion, it is obvious that 



O.O.U=(DxO)U + (n*0)U, 



O.X2.U=(HxO)U + (0*)U; 



N2 



