to the Cartesian Ovals in piano. 



the foci of the conic U + W will be given by the equality 

 « + \a h-t-Xrj y + X-7 1 



383 



0, 



h + \rj b + \@ /+X$ i 



ff+7vy f+\(f> -■(•+*) 



1 i — (x + iy) 



which, by equating real and imaginary parts, gives two equations 

 between x, y, \. 



By aid of these equations \ may be expressed as a rational 

 integral function of x, y, which we know a priori must be of the 

 second degree only, since otherwise, on substituting for its value 

 in either of the equations between x, y, \, we should obtain an 

 equation above the sixth degree in x, y, contrary to Chasles's 

 theorem (we shall also see that this is the case, without having 

 recourse to this theorem, by the reasoning below) . 



Let # 2 -f-y 2 = 0, then the above equality becomes 



showing that the origin, i. e. any one at will and therefore all 

 of the four fixed points, is a focus. If X were above the second 

 degree in x, y, the line joining this point with either of the 

 circular points of infinity would be always at least a triple tan- 

 gent to the focal curve ; but in the case where the focal curve 

 breaks up into two distinct subloci, we have seen that these tan- 

 gents to each sublocus are simple, i. e. that they are double in 

 regard to the whole locus, wherefore X must be always a qua- 

 dratic function only of x, y. Consequently each circular point 

 of infinity must itself be a double point upon the curve. 



If now we regard the four given points as syzygetic foci of a 

 curve (a term indispensable to give precision to the theory of 

 foci when interpreted in the Pliiekerian sense), s. e. if we sup- 

 pose a plane curve defined by the equation 



\ a/A + fjb v/B + v \/C + 7tv/D= 0, 



where A, B, C, D are the characteristics of the infinitesimal 

 circles of which the four given points are the centres ; and if 

 in the norm of the linear function above written we make 

 \ + [j, + v + 7r = 0, and conjoin with this two other equations be- 

 tween X, /J,, v, ir, which will make the term (<r 2 + y 2 ) 3 (Lx + My) 

 in the norm vanish identically, i. e. the equations L = 0, M = 0, 

 we shall obtain a group* of curves of the sixth degree, each pos- 

 sessing precisely the same geometrical characters as have been 

 proved to be satisfied by the curve of foci of the conies U + XV 

 drawn through the four fixed points, viz. of having the circular 



* No two of the group can be identical; for in such case one of the focal 

 distances could be eliminated, and the syzygy be reducible by one term, 

 contrary to hypothesis. 



