the Dialytic Method of Elimination, 225 



functions employed may be, I shall for the sake of greater sim- 

 plicity restrict the demonstration to the case of three functions, 

 U, V, W, whose degrees (if unequal, written in ascending order 

 of magnitude) are m, n, p respectively. Let 



V:=zYm{oo, 2/) +Fm-t(<r, y)z' 



Y^Gn{x,y) + On-i[x,y)z'- 



W = Yip{x,y) + np-,[x, yy- 



Let p, q be taken any two numbers which satisfy in integers 

 greater than zero the equation 6 -\-(i) = m-\-\, and let 



^m{x,y) = <f>m-Q.X^ + (f>m-u).y'^ 



^n[x, y) = 7n-0 . X^ + yn-oy.y'" 



}ip{x,y)=7]p-9.X^-^rjp-a>.y'^, 

 where the <^'s, 7's, rj's may be always considered rational integer 

 functions of x and y ; for every term in each of the functions 

 F, G, H must either contain x^ or y<^, since, if not, its dimen- 

 sions in X and y would not exceed (0—1) + {(0—1), i. e. m — 1, 

 whereas each term is of m conjoined dimensions, at least in x 

 and y. Hence from the equations 



U = 

 V=0 

 W=0, 

 by eliminating x^, y^ we obtain the connective determinant 



yn—e; 7w— w; Gn—i 

 Vp-O'f Vp-<o; lip-i, 

 which will be of the degree 



(m + 7i4-jo-(^ + tw + 0). 

 i. e. of the degree [n+p—t — l] in x and y ; and the number of 

 such connectives by principle (c) is p. 



Again, by augmentation we can raise each of the functions 

 U, V, W to the same degree as the connectives, and by principle 

 b the number of such will be 



n-\-p—m—i 

 p-i 

 n — v 

 from U, V, W respectively, together making up the number 

 2ri + 2jt?--m— 3t. 

 Hence in all we have %n + 2p — 3t equations ; and the number 



