314 Dr Lachlan, On the degree of the [May 24, 



usually be coincident with the vertices of the triangle. Excluding 

 all these points let us suppose that there are r points of inter- 

 section which do not lie on the lines x, y, z\ and let cj) r (x, y) = 

 be the equation which represents the line connecting these points 

 to the point x = 0, y = 0. 



Then (j> r (x, y) = is what is usually called the ^-eliminant of 

 the equations 



f m {oc,y,z) = 0, f n (x,y,z) = 0. 



2. It is obvious from this method of forming the eliminant 

 of two equations, that, whichever one of the three variables be 

 eliminated from two homogeneous equations, the result is of the 

 same degree. 



This fact may often be used with advantage. 



As an example, suppose we require the degree of the eliminant 

 when x is eliminated from two equations of the form 



u m + \u p = 0, 



u n + \u q = 0, 



where u m , u p , u n , u q denote functions of x, y of degrees m, p, n, q. 



Here if \ be eliminated the eliminant is 



and is of degree equal to the greater of the two numbers m + q 

 or n +p. 



Hence, if x be eliminated from the two equations the eliminant 

 will contain \ in the degree m + q or n + p, whichever is the 

 greater. 



Similarly, it follows that the discriminant of 



will contain \ in the degree m+p — 1. 



3. The degree of the eliminant of any two equations is to be 

 found by arranging the equations in a homogeneous form, intro- 

 ducing if necessary other variables. The equations are then to 

 be considered as representing curves, and the number of points 

 of intersection which lie on the lines forming the triangle of 

 reference has to be determined. The degree of the eliminant is 

 equal to the product of the orders of the curves diminished by 

 the number just determined. 



