PEOFESSOE CAYLEY ON CUEYATUEE AND OETHOGONAL SURFACES. 
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such that it meets the tangent plane in the sibicon jugate 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 
((A),..OCa,..)=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. 
10. The principal cone, writing, for shortness, 
h%+h+j%, g^+fi+c^=^, h, 
was before taken to be the cone 
I, * , K 
X, Y, Z 
Representing this equation by 
i(A, B, C, F, G, HX§, £) 2 = 0, 
the expressions of the coefficients are 
A=2AZ — 2</Y, 
B =2/X— 2AZ, 
C =2y Y— 2/X, 
F = h Y — gZ -(b-c)X, 
G=fZ- hX-(c^-a) Y, 
FI= gX- fY-( a -b)Z. 
These values give 
AX+FIY+GZ=Z£Y — Yc$Z, 
HX+BY+FZ=XSZ -ZIX, 
GX+FY+CZ=Y&X-X&Y; 
whence also 
(A, ...JX, Y, Z) 2 =0, 
as is, in fact, at once obvious from the determinant-form ; and also 
A+B+C=0. 
17. Writing, for shortness, 
(a, b, c, J\ g, h) — (bc—f 3 , ca—g\ cib—h 2 , gh — af, hf~bg,fg—ch), 
we find 
Aa+m+Gg= a (hZ — gY)-\-hZ —gX, 
m+Bb+Yf= a (fX-hZ )+fX-hZ, 
G g + Ff+Cc=4 9 Y -fX)+~gY-JX ; 
