438 On the Theorem that every Algel)rnic Equation has a Root. 



cident lines, and there would be in all more than >i lines of in- 

 tersection of the plane and cone. 

 Consider now the equation 



(f)u = 0, 



where <f>u is a rational and integral function of u with (in general) 

 imaginary coefficients, and write 



r, Q being real functions of {x, y), each of them of the degree 

 n; if {x,y) are rectangular coordinates, then P = 0, Q=0 

 are real curves each of the order n. And to each point of inter- 

 section of the two curves there corresponds a root of the equa- 

 tion. The two curves do not intersect in more than n points 

 (for if they did, the equation <f>u = would have more than n 

 roots) ; hence if it be shown that the two curves intersect in at 

 least n points, they will intersect in precisely n points, and the 

 equation will have 7i roots. Take any point as the common ver- 

 tex of two cones standing upon the curves P=0, Q=0 respec- 

 tively ; each point of intersection of the two curves corresponds 

 to a line of intersection of the two cones, and it is only necessary 

 to show that the two cones intersect in at least n lines. Take 

 for the vertex a point in the perpendicular at the origin of {x, y) 

 to the plane of the two curves, and at a distance unity from such 

 origin, viz. a point such that, treating it as the origin of the 

 coordinates [x, y, z), the coordinates in respect thereto of the 

 origin [x, y) are x = 0, y=0, z:=l. The equations of the cones 

 are at once deduced from those of the curves by writing therein 



I -, - j in the place of (a?, y) and, to render the equation integral, 



multiplying by s'\ Or if P' = 0, Q' = are the equations of the 

 cones, we have 



Consider the section by the plane through the vertex parallel 

 to the plane of the two curves : the equation of this plane is 

 z = ; and it is clear that, to obtain the intersections of this cone 



with the plane in question, we have only in (pi '^ ^ — ~ 1 to 

 disregard all the terms after the first. Suppose that 



(f)U={a + b \/ — 1 )m" + &c. ; 

 then putting 



{a + b ^/~l){x + y ^~l)» = P'o-l-Q'o v/'^l, 



the equations P'o = 0, Q'o=0 determine the intersections of the 



