Prof. Cayley on Differential Equations and Umbilici. 449 

 for which the derived equation in its complete form is 



(2x-yp) 2 -(2x 2 -y 2 )=0, 



or, what is the same thing, y 2 p 2 — 4xyp + 2x 2 + y 2 = 0, and for 

 which therefore there is no factor to divide out. 



VII. 



The conies confocal with a given conic form a system similar 

 in its properties to that of the curves of curvature of a quadric 

 surface ; and the theory of the last-mentioned system may be 

 studied by means of the system of confocal conies. Consider 

 then the equation 



. j£ I 



a 2 + z ' b 2 + z 



which, if z be an arbitrary parameter, belongs to the conies 



x v 

 confocal with the ellipse -g + jg = 1. Treating z as a coordinate, 



the equation represents a surface of the third order, which is such 

 that its section by any plane parallel to the plane of xy is a conic ; 

 and the confocal conies are the projections on the plane of xy, by 

 lines parallel to the axis of z, of the sections of the surface. 



The sections by the planes of zx, zy are the parabolas x 2 =z + a? 

 and y 2 = z-\-b 2 respectively. When z>—b 2 , the ordinates in 

 each parabola are real, and these ordinates give the semiaxes of 

 the elliptic section. When z>— a 2 < — b 2 , then only the para- 

 bola section in the plane of zx has a real ordinate, and the sections 

 are hyperbolic; and when z< — a 2 , the section is altogether 

 imaginary. The section in the planes z~ — b 2 is the pair of coin- 

 cident lines ?/ 2 =0, z— — b 2 , and the section in the plane z=—a 2 

 is the pair of coincident lines z= — a 2 , x 2 =0 ; or, in other words, 

 the plane z-\-b 2 =0 touches the surface along the line ?/ = 0, and 

 the plane z + a 2 =Q touches the surface along the line x=0: this 

 at once appears from the integral form 



(z + a 2 ){z + b 2 )-x 2 (z + b 2 )-y 2 {z+a 2 )=0. 



The poi nts (z- -b 2 , y=Q, a?= ±\/a 2 -b 2 )and (z= -a 2 ,x=0, 

 y= ±\Zb 2 —a 2 ) are conical points; the last two are however 

 imaginary points on the surface. To find the nature of the sur- 

 face about one of the first-mentioned two points, say the point 

 (z = — b 2 ,y = 0, x = x/a 2 — b 2 ), taking this point for the origin and 

 writing therefore \/ a 2 — b 2 + x, y, and — b 2 + z in the place of 

 x, y, z respectively, the equation becomes 



(a 9 --b 2 + z)z-((a 2 -b 2 )+2x\/¥^ 2 + x 2 ')z-(a 2 --b 2 + z)y 2 = 0, 



