434 Prof. Sylvester on a linear Method of Eliminating hetixieen 



but, now since a + /3 + y = » + 2, we shall easily see by the 

 same method as above, that the least value of « + 6 +c 

 (where a, h, c denote respectively the greatest values of a, /3, y, 

 appearing in the denominator of the fractionalyb;7«s used to 

 express II (a, /3, y), will be one greater than before, or n ; so 

 that f-\-g + h + a-\-b + c will still be equal to 3« — 2, as 

 we might, a. priori, by virtue of our rule, have been assured. 



Ternary Systems. 

 Case (B). — Two of the indices equal ; the third less by a tinit. 



Let U = 0, V = 0, W = 0, be the three given equations se- 

 verally of the degree n,n,{7i — \). 



Make ?• + ;•' + ?•" = « — 2, s -f s' + s" = « — 2, t + t' + t" = n — l, 



by multiplying U into x^ .y'^ . z^' , V into x^ .1/ . z" ", W into 



j:' . y . z' , we obtain augmentees each of the same, namely, 



the (27i — 2)th degree. 



rp, 1 r ^u • (n — \\n {n — \\n n .n -^-X 

 Ihe number ot these is, ^^ '- 1- ^^ '- — 1 — . 



2 2 2 



Again, make «4-/3 + y = 7« + l. 



It will still be possible, as before, to form equations of de- 

 composition in which a-" rf" z^ are the arguments, and affected 

 with integer factors. For if we look to W even, all its argu- 

 ments are of the form r" . ^ . z'^, where a -\- b + c = [n — \), 

 and each of these cannot be less than its correspondent, for that 

 would be to say that (>« — 1 ) is not greater (?< + 1 ) — S, a fortiori, 

 U and V can be decomposed in the manner described. 

 Thus, then, we shall obtain as many secondary derivees as in 



the last case (Method 1.), i. e. ' (since a + /3 H- y is 



still equal to n + l), as before. Moreover, each of these will 

 be of {n — a.) + («— /3) + {n—i—y), i. e. of 2 « — 2 dimen- 

 sions. 



Altogether, therefore, we have 



{{n — \)n (ii — \)n n{n + \)\ {n — \).n 

 2 + 2 -^ 2 /+ 2 



linear independent equations of the degree 2 ?? — 2, and the 



(2 ;^ 1 ) 2 71 



number of arguments to eliminate is ^^ — -^ . Now 



these two numbers are equal. Thus we obtain a final deri- 



. . - TT' fC • , (" — 1)« . (" — 1) " 



vee containmg 01 U s coeihcients ^^ — ^ 1- ^^ ^^— ^ — , an 



