1897.] Eliminant of Two Algebraic Equations. 317 



This is obvious by eliminating f from 



f a?i fen _ , 





i&c. 4. If # be eliminated from the equations 



+ ; f^ +•••+/ ^^^ = 0, 



(a, + #f (as + ey T ' • • T (a n + 0)* 



+ ; — r3xs + — + 7 — rms = 0, 



fo + tf)* (o 2 + ^)« "" K + 0) 9 



the eliminant is of degree (n — 1) p — q (p > q). 



Introducing f and rj to make the equations homogeneous, we 

 may write them in the form 



& + _^ +...+ VXn = 0, 



(a lV + ey (a^ + ey (a nV + o)p 



A X %X X A 2 7)X 2 A n rjX n _ 



(a lV + ey (a 2V + ey ' (a nV + ey 



If now we eliminate £ we get 



1 [ A 2 x 2 A n x.„ 



+ .,. + ^_ 



(a lV + ey {(a 2V + 0)p '" (a nV + 0)*) 



If p > q, this result is of the (n — 1) p — q degree with respect 

 to 7) and e. 



Hence by § 2 the result of eliminating e from the equations 

 will be of degree (n — l)p — q with respect to £ and ??. 



Ex. 5. If x be eliminated from the equations 



a x mi + ... + arX°' m - r + ... + a r 'x 2 + ... +a ' = 0, 

 box™ + ... + brX m ~ r + ... + b r 'a? + . . . + W = ; 

 where a r , a/, b r , 6/ contain y in the rth degree, the eliminant is 

 of degree %nn. 



Consider the curves 



C 2m = UzX™ + ...+ llrX 2 ™-* +...+ Ur'nfZ*"^ +...+ U*Z mx = 0, 



G m = v x m + ...+ v r x m ~ r + ... + v/x^-v + ... + v 'z- n = ; 



where u r , u r ', v r , v r ' are homogeneous with respect to z and y of 

 degree r. 



