128 
ON DIFFERENTIAL EQUATIONS AND UMBILICI. 
[330 
VII. 
The conics 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 conics. Consider 
then the equation 
-£-+-£.«1 
a 2 + z b 2 + z 
which, if z be an arbitrary parameter, belongs to the conics confocal with the ellipse 
— + ^ = 1. Treating 2 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 conics 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 2 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 parabola 
section in the plane of zx has a real ordinate, and the sections are hyperbolic; and 
when z < — ct 2 , the section is altogether imaginary. The section in the planes z = — b 2 
is the pair of coincident lines y 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 y = 0, and the plane z + a 2 = 0 touches the surface 
along the line x = 0: this at once appears from the integral form 
(z + a 2 ) (z + b 2 )-x i (z-\- b 2 ) — y 2 (z + a 2 ) = 0. 
The points (z = — b 2 , y = 0, x = ± Va 2 — b 2 ) and (z = — a 2 , x = 0, y = ±Vb 2 — a 2 ) are conical 
points; the last two are however imaginary points on the surface. To find the nature 
of the surface about one of the first-mentioned two points, say the point (z = — b 2 , 
y = 0, x = ^Ia 2 — b 2 ), taking this point for the origin and writing therefore Va 2 — b 2 + x, y 
and — b 2 + z in the place of x, y, z respectively, the equation becomes 
(ia 2 — b 2 + z)z— ((a 2 — b 2 ) + 2x Va 2 — b 2 -f- x 2 ) z — (a 2 — b 2 + z)y 2 = 0, 
that is 
z 2 — 2zx Va 2 — b 2 — (a 2 — b 2 ) y 2 — z (x 2 + y 2 ) = 0 ; 
so that there is a tangent cone the equation whereof is 
z 2 — 2zx Va 2 — b 2 — (a 2 — b 2 ) y 2 = 0, 
or, as it may be written, 
(z — x'Ja 2 — b 2 ) 2 — (a 2 — b 2 ) (x 2 + y 2 ) = 0.