238 PROFESSOR CAYLEY ON CURVATURE AND ORTHOGONAL SURFACES. 
or, what is the same thing, 
X 7 — X( 1 -f- d^a + d y p -f- d z py) — Xd^ge c — Yd a .p — Z d^y, 
with the like expressions for Y' and 71. 1 proceed to reduce these. The formula for X' is 
X'=X{1 + §(X a + dp -f- d z y) -f- ad,o -\-ftdyp yd z % } 
- qiXdji+Ydp + Zdjy) - («X+0Y+yZR§. 
23. I write, for shortness, c> = X$ r Y d ;J -j- d , , whence SX, &Y, SZ=«X + /iY+^Z, 
AX + 0Y+/Z, ^X+/Y+cZ, agreeing with the former significations of SX, &Y, dZ ; 
also Yd x Y, Yd y Y, Y d z Y hY, hZ, and Y&V=X&X+YSY-j-Z&Z. It is now easy to 
form the values of 
d,a, dp, d x y. 
viz. these are y— 
xsx 
V s ’ 
4 
Y 
YSX 
V 3 ”’ 
g 
V 
zsx 
v 3 ’ 
dyCL , dp, d 3 y , 
h 
Y 
XSY 
1 v 3 5 
6 
V 
YW 
— Y 3 ~’ 
/ 
V 
ZSY 
V 3 5 
d/x, d-P, d.y, 
9 
V - 
xsz 
’XX ’ 
/ 
V 
YSZ 
- v 3 ’ 
c 
Y~ 
zsz. 
- y-3 ’ 
and hence 
7 , 7 a , 7 « + 5 + c 2V 
d,a + T/i + d z y — — y y S ’ 
IX V 2 
X<U+Ydj3+Z<Z.y=^-f 8 &X, =0, 
<X+0Y + yZ =Y; 
and we have 
x'=x{i +f (X+f_^ +k ? }-V3,f, 
with the like values of Y' and 71. But we are only concerned with the ratios X : Y' : 7J ; 
whence, dividing the foregoing values by the coefficient in { }, and taking the second terms 
only to the first order in g, w 7 e have simply 
X, Y', Z'=X-Yd,g, Y -Yd„q, Z-V<4g. 
24. We may investigate the condition in order that the surface x', y', z' may be the 
consecutive surface r-\-dr=r(x, y, z). This will be the case of 
r+dr=r(i i’+gy, y+%\ s+g |), 
that is, r+dr=r+gY, or g=y. This value of g gives d x ^=—~ 2 d x V= — Y^'K, and 
similarly d y § = — yXY, d^=— yXZ; whence 
X, Y', Z'=X+-^X, Y+fXY, Z + |^Z, 
