1897.] Eliminant of Two Algebraic Equations. 315 



To illustrate the method let us take the example given by 

 Serret. He takes the equations 



0, 2) y* + (x, 2) f + (x, 4) y 2 + (x, 5)y + (x, 5) = 0, 



O, 8) f + (x, 6)i/* + (x, 9) f + (x, 4) f + (x, S)y + (x, 4) = 0, 



where (x, r) denotes an integral expression of the rth degree in x. 



Introducing z, so as to make these equations homogeneous, we 

 have the equations of two curves, C 6 , G 13 ; 



(x, z\ if + {x, z\ y*z + (x, z\ if + (x, z) 5 y + (x, z\ z = 0, 



(x, z\ if + (x, z\ y'z s + (x, z\ y 3 z + (x, z\ ifz 7 + (x, z\ yz 9 + (x, z\ z 9 = 0, 



where (x, z) r denotes a homogeneous expression of the rth degree. 



.The only common points of the curves C 6 , C 13 which lie on the 

 lines x, y, z coincide with the points x = z = 0, and y = z = 0. 



In fact, at the point x = z = 0, 



C 6 has two branches, given by equating the coefficient of y* to 

 zero, 



G 13 has eight branches, given by equating the coefficient of y 5 

 to zero. 



Hence at this point there are 2x8 points of intersection of 

 the curves. 



Again, at the point y = z = 0, 



G 6 has one branch, given by equating the coefficient of x 5 to 

 zero, 



C 13 has two branches of the form 



y 2 + axz = and y 3 x 5 + 6z 8 = 0. 



Hence at this point there are 4 points of intersection of the 

 curves. 



Hence the order of the eliminant of the equations 



= 6x13-2x8-4 = 58, 



which agrees with Serret's result. 



4. The following examples are of some importance. 

 Ex. 1. If 6 be eliminated from the equations 



T-7T+ ?-» + ». + 7-71*1, 



a x + a 2 + 6 a n + 



+ ZT~Ta + • • • + z — Ta = X > 



a x + 6 a 2 +0 a n + 



the degree of the eliminant is n — 1 . 



