PROFESSOR CAYLEY ON CURVATURE AND ORTHOGONAL SURFACES. 245 
+ V( - Y dj, + Zd/l+Xd; -Xd% 
G" = (V7»-^A) i 4 +- V j(c - a) (ZSZ - XSX) jf ?,5 - (v/- ™) d 4 
+ V( - Zd,d,+Xd,d, + Ydl - Y <5)8, 
H , '=-(’'?-T^)4?-(v/-A' r )^ + |v(«-J)-i(XSX-Y5Y)k ? 
-f- V( — \d z d x -f- Y d z d y -f- 7id x — Z d y )q. 
35. It will be recollected that we have X'l'+YY/ffiZ'^^XI+Y^-J-Z^ ; by what 
precedes it appears that for the given surface the principal tangents are determined by 
the equations 
(A, . .11, *, ff= 0, 
Xg-f-Y t?-TZ£=0, 
and that the lines which (in the tangent plane of the given surface) correspond to the 
principal tangents of the corresponding point of the vicinal surface are determined by 
the equations 
(A, . .XI, ,, £) 2 + (A", • •!?, fl, £) 2 = 0, 
Xg+Y fl +Z?=0. 
Condition that the two surfaces may belong to an Orthogonal System. 
Article Nos. 36 to 41. 
36. The condition in order that the two surfaces may belong to an orthogonal system 
is that the principal tangents shall correspond, or, what is the came thing, the lines which 
(in the tangent plane of the given surface) correspond to the principal tangents of the 
vicinal surface must be the principal tangents of the given surface. When this is 
the case the plane and cone X|+Y r /j + Z£=0, (A", . . .X§, *7, £) 2 — ^ intersect in the 
principal tangents, and this is therefore the required condition. 
The plane X|-j-Y^+Z^=0 meets the cone (A", . .fz, q, = 0 in the principal 
tangents, that is in a pair of lines harmonically related to the circular lines and also to 
the chief tangents. Forming then the coefficients (A"), (B"), (C"), (F"), (G"), (Ft") 
from A", &c. in the same way as (A) &c. are formed from A, &c., that is, writing 
(A") = B"Z 2 -f- C"Y 2 — 2F" YZ, &c., the conditions are 
(A") + ( B") -j- (C") = 0, 
((A"), . . .fa, . . .) = 0, or, what is the same thing, 
(A", . . . X(4 • • 0=0. 
The former of these, as about to be shown, is satisfied identically ; we have therefore 
the second of them, say (A", . . .) = 0 as the required condition. 
MDCCCLXXIIT. 2 L 
