PROFESSOR CAYLEY ON CURVATURE AND ORTHOGONAL SURFACES. 237 
Forming the sum PX -j- QY+ RZ, the coefficient of £X is found to be 
= -Z(/iX+6Y+/Z)+Y(^X+/Y+cZ), = — Z&Y-j-YSZ ; 
hence the whole is 
=&X(Y&Z-Z5Y)+&Y(ZSX— X&Z)+SZ(X&Y-YSX), which is =0, that is, 
PX+QY+RZ=0. 
21. Hence, adding, we find 
((A),... !«,..) = 0; 
^ iz. in this and the before-mentioned equation 
(A)+(B)+(C)=0 
we have the ci posteriori verification that the cone (A, . . .ff, r h £) 2 =0 cuts the tangent 
plane in the double lines of the involution. 
In what precedes I have given only those relations between the several sets of quan- 
tities a, a, ( a ), A, (A), &c. which have been required for establishing the results last 
obtained; but there are various other relations required in the sequel, and which will 
be obtained as they are wanted. 
The Conormal Correspondence of Vicinal Surfaces. Article Nos. 22 to 35. 
22. We consider a surface U = 0 (or r=r), and at each point P thereof measure 
along the normal an infinitesimal length g, dependent on the position of the point P 
(that is, § is a function of x, y, z). We have thus a point P', the coordinates of which are 
x J , y', z'=x-\-oa, y-f§3, z + qy, 
where a, 3, y are the cosine-inclinations of the normal, that is, 
if v=ys>+r+z>; 
the locus of P' is of course a surface, say the vicinal surface, and we require to find 
the direction of the normal at P', or, what is the same thing, the differential equation 
XI dx' -f Y'dy' -j- r ZJdz' of the surface. We have 
dx = (1 ~\~d x qcf dx- f- d^qa . dy -f- d z qc *. . dz , 
dy 1 — d.,ofz . dx (1 -J- ^§3) dy~\~ d e gfi . dz, 
dz'= d,oy dxAr d y q y .dy+( 1 + d z q y ) dz, 
0 = X dx -f- Y dy-\- Z dz; 
whence, eliminating dx, dy, dz, we have between dx', dy', dz' a linear equation, the 
coefficients of which may be taken to be X', Y', Z'. Taking these only as far as the 
first power of g, we have 
X' =X(1 + fqfi -j- d z qy) — Yd x o$ — Zd x qy, 
2 K 
MDCCCLXXIII. 
