Fundamental Theorem of Invariants. 183 



D(w : i,j) = A(w : i,j) must be satisfied from the maximum 

 value of w down to the value 0, which proves the great hitherto 

 undemonstrated fundamental theorem for a single quantic. 



For any number of qualities the demonstration is precisely 

 similar at all points : there will be as many systems of i, j as 

 there are quantics. (w : i,j : i',f : &c.) will denote the num- 

 ber of ways of making up w with j of the integers 0, 1, 2, . . . i, 

 with/ of the integers 0, 1, 2, . . . i f , and so on. The theorem 

 to be demonstrated will be 



D(w: i,j: i',f : ...) = A(w: i, j : %' :/...)• 



CI will become 2 ( a -^ + 26 j- + ...J, 



o „ „ *(»^+(*-i>| +...). 



It w T ill still be true that II 9 . ? . D, where D is a differentiant 

 in x (i. e. a function of the elements in all the given quantics 

 which withstand change when these are transformed by writing 

 x + hy for x), is a numerical multiple of D; and D will be sub- 

 ject to the identity HD = 0. We shall still have 



T>(w: i,j: i',f :...)= or > A(w : i,j : i',f : ....), 

 and 



D(_0:i,j:i',f:...) = (0;i,j : if, f :...), 

 and shall be able in precisely the same way as before to de- 

 monstrate the impossibility of SjzjJ, ~D(w — k : i,j : i', f) being 

 greater than (iy : i, j : i f , f : . . . ), and so shall be able to infer by 

 the same logical scheme A(w :i,j: i' ,f :...) = D(w : i,j : i r , f) 

 my extension of Professor Cayley's theorem, which leads 

 direct to the Generating Fractions given in my recent papers 

 in the Comptes Rendus. 



In a series of articles which I hope to publish in the Ame- 

 rican Journal of Pure and Applied Mathematics, I propose to 

 give a systematic development of the Calculus of Invariants, 

 taking a differentiant as the primordial germ or unit. I have 

 spoken of a differentiant in x, and of course might have done 

 so equally of a differentiant in y. If we call the former D x , 

 it is capable of being shown, from the very natures of the 

 forms and £1, that if the quantity ij — 2iv, which may be 

 called the degree of D x , be called 8, then 5 Da, becomes a dif- 

 ferentiant in y. These may be termed simple differentials ; 

 but the principle of continuity forbids that we should omit to 

 comprise in the same scheme the intermediate forms O^D^. or 

 £l q T) y , through which simple differentiants in x and y pass 

 into each other. These may be termed mixed differentiants ; 



