182 Prof. J. J. Sylvester's Proof of the undemonstrated 



> is not the sign to be employed, then this equation, for every 

 value of iv, commencing with the assumed one down to 0, 

 must be satisfied. The greatest value of w for given values of 



i 9 j, it is well known, is ^ for ij even, and ^ — for ij odd. 



Let us give to w this maximum value in the above " greater 

 or equal" relation ; for brevity, denote the differentials whose 

 types kre[w,i,j~\, [w— l,i,f\ ...by [>], [w — 1], [w— 2], &c 

 respectively, i and j being regarded as constants. It will be 

 convenient to substitute for the number of arbitrary constants 

 in any of these differentiants the same number of linearly in- 

 dependent specific values of them; so that we shall have 

 D(iv :i,j) of linearly independent [w]% D(iy — 1 :i,j) of 

 linearly independent \iv — l]'s, and so on. Now, instead of 

 the D(iv — q : i, j) differentiants [w — q~\, let us substitute the 

 same number of the derived forms (O q [w — 5'])'s. I shall prove 

 that the quantities (all of the same iveight w) thus obtained 

 are linearly independent of one another. 



For (1) suppose that those belonging to any one set 

 O q . [w — q~\ are not independent, but are connected by a 

 linear equation. Then, operating upon this equation with H ? . , 

 we shall obtain a linear equation between the quantities [w — q~] 

 for each quantity (fl g . O q . \io — q~\ being a numerical multiple of 

 \_w — <7]), which is contrary to the hypothesis. Again, let there 

 be a linear equation between the quantities contained in any 

 number of sets of the form q . \iv — q~\ for which m is the 

 greatest value of q. Then, operating upon this with Dt w , it is 

 clear that all the quantities in the sets for which q < m will 

 introduce quantities of the form £l m ~ q \_w — q~] where m—q>0 } 

 and which consequently vanish. There will be left, therefore, 

 only quantities of the form [iv — q~\, between which a linear 

 equation would exist, contrary to hypothesis, as in the pre- 

 ceding case. Therefore all the quantities in all the sets 

 are linearly independent. But these are all of the weight iv, 



i. e. hj or - 9 ,and are therefore linear functions of the num- 

 ber of ways in which the integers 0, 1, 2, 3, . . . % can be com- 

 bined % and j together so as to give the weight w. Therefore 

 being linearly independent, as just proved, their number can- 

 not exceed this last-named number, i. e. cannot exceed (w : ?,/). 

 That is to say, 



DO : i, jO + DO-1 : i,j) +... +D(0 : i,j) 

 cannot exceed (w : i, j). Therefore every one of the equations 



