298 
ON CURVATURE AND ORTHOGONAL SURFACES. 
[519 
15. I take r — r = 0 for the equation of a surface, X, Y, Z for the first derived 
functions of r, (a, b, c, f, g, h) for the second derived functions. The equation of the 
tangent plane at the point (x, y, z), taking f, y, £ as current coordinates measured 
from this point, is 
XIt + Yvj + Z£ — 0 ; 
the equation of the chief cone in regard to this form of the equation of the surface is 
(a, b, c, f, g, y, £) 2 = 0, 
and the equation of the circular cone is f- + rf + = 0, or, what is the same thing, 
(1, 1, 1, 0, 0, 0■$£, y, £) 2 = 0. 
Imagine a quadric cone 
{A, B, C, F, G, y, £) 2 = 0, 
such that it meets the tangent plane in the sibiconjugate lines of the involution 
formed by the intersections of the tangent plane by the chief cone and the circular 
cone respectively; that is, in a pair of lines harmonically related to the intersections 
with the chief cone, and also to the intersections with the circular cone; the 
conditions are 
((4)....0, 
and 
(A) + (B) + (C) = 0, 
viz. if only these two conditions are satisfied the cone will intersect the tangent plane 
in the two principal tangents. 
16. The principal cone, writing, for shortness, 
+ hy + g£ hi; + by +/£ pf +fy + c£ = 8%, 8y, 8f, 
was before taken to be the cone 
Ç, 
v > 
? 
8y, 
8Ç 
X, 
Y, 
Z 
Representing this equation by 
HA, B, C, F, G, y, ?) 2 = 0, 
the expressions of the coefficients are 
A = 2hZ - 2gY, 
B = 2fX — 2 hZ, 
G = 2gY- 2/X, 
F = hY — gZ — (b — c) X, 
G = fZ - JiX-(c-a) Y, 
11= gX- fY — (a — b) Z.