234 PROFESSOR CAYLEY ON CURVATURE AND ORTHOGONAL SURFACES. 
Writing also 
aX + hY+g Z, hX+bY+fZ, gX+fY+cZ=%X, SY, $Z, 
and X 2 -f-Y 2 + Z 2 =V 2 ; also a-\-b-\-c=co , then 
(a) = (b + c)Y 2 -*X 2 +XSX-YSY-ZSZ, 
(b) =(c+a)V 2 -v Y 2 — XSX + Y<5Y — Z&Z, 
(c) =(a+b)V 2 -cvZ 2 -XbX-YbY + ZbZ, 
(/)=— /Y 2 -FFZ+YSZ +ZSY, 
(^) = -(/V 2 — &ZX+Z&X +X£Z, 
(7i) = -hY 2 — &YZ+XSY+YSX. 
14. I give also the following lemma: — 
Lemma. The condition in order that the plane X|-J-Y?j-|-Z^=rO may meet the cones 
(A,B,C,F, G, HXi, 9 , ?) 2 = 0, 
(A', B', C', F, G', H'X5, 9, Q 2 =0 
in two pairs of lines harmonically related to each other, is 
(BC'+B'C — 2FF, . . , GH' + G'H-AF-A'F, . . JX, Y, Z) 2 =0. 
Writing here 
(A, . . 1[Y%—Z?i, Zg-X£, x«-Yi) 2 
=((A), (B), (C), (F), (G), (H)XI, », 
that is, (A)=BZ 2 +CY 2 — 2FYZ, &c., the condition may be written 
(A) A' + (B)B' + (C)C' + 2(F)F + 2(G)G' + 2(H)H' = 0, 
or say 
((A), . . Jk', ..)=«; 
and we may, it is clear, interchange the accented and unaccented letters respectively. 
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 g, >?, £ as current coordinates measured from this point, is 
Xg+Y9+Z£=0; 
the equation of the chief cone in regard to this form ot the equation of the surface is 
0, b, c,f ; g, 9, ?) 2 =0, 
and the equation of the circular cone is | 2 +?f + £ 2 =0, or, what is the same thing, 
(1, i, i, o, o, ox?, 9, ?) 3 = 0. 
Imagine a quadric cone, 
(A, B, C, F, G, FI XI, 9, £) 2 =0, 
