PROFESSOR CAYLEY ON CURVATURE AND ORTHOGONAL SURFACES. 233 
then, to determine the coordinates (u lt v„ w r ), (u 2 , v 2 , w 2 ) of the points of intersection of 
the line and conic, we have 
(a, . . 3 [Y£ — Zjj, zg-X£, X^-Y?) 2 
= + + ?W,)(|m 2 + fjv 2 + £w 2 ), 
or, what is the same thing, we have 
(a, . . . XY£— Z? 7 , Zg-X?, X^-Y?) 2 =0 
as the equation, in line coordinates, of the two points of intersection. The proof is 
obvious. 
12. Making the equations refer to a plane and a cone, and writing throughout >?, t 
as current point coordinates, the theorem is : — 
Given the plane Xg+Y^-f-Z^=0, and cone 
(a, b, c,f ; g, hXZ, n, £) 2 =0 ; 
then, to determine the lines of intersection of the plane and cone, we have 
(a, • • XY^-Z^, Zg-Xf, X^-Yg) 2 =0 
as the equation of the pair of planes at right angles to the two lines respectively. 
13. Denoting the coefficients by (a), ( b ), &c., that is, writing 
(a, . . XY^ — Z?;, Z' — X£, Xjj — Y?) 2 
=((«), (J), (c), (/), (<?), (A)X5, ,, O', 
the values of these are 
(a) = bZr +cY 2 — 2/*YZ, 
(A) = cX 2 +«Z 2 — 2^ZX, 
(c) = «Y 2 +AX 2 -2AXY, 
(/) = — «YZ — /X 2 +^XY+AXZ, 
(7/) = — AZX +/ YX — ^Y 2 + AYZ, 
(A) =-cXY+/ZX+^ZY-AZ 2 . 
We have the following identities : — 
(«)X+(A) Y+fo) Z=0, 
(A)X+(A)Y+(/)Z=0, 
fo)X + (/)Y+(c) Z=0, 
• • , (f/m-W), ■ ■ -)=-(X 2 , Y 2 , Z\ YZ, ZX, XY)<p, 
that is, (£)(c)— (/)'=— Xhp &c., where 
$ = (/«-/', . . y/t-tt/, . . IX, Y, Z)». 
