PROFESSOR CAYLEY ON CURVATURE AND ORTHOGONAL SURFACES. 
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which, is as it should be, viz. these are what X, Y, Z become on substituting therein for 
x, y, z the values y-\~^ 3, z-\-q 7 . 
25. I return to the case where § is arbitrary, and I investigate the values of a, b, . . . 
for the point P' on the vicinal surface; say these are a!, b', See., then we have d=dJX! Sec. 
The relation between the differentials may be written 
dx=(l — d x oa)dd — dyOa dy ' — d,oa dd , 
dy— —d x $(5 dd + ( 1 — d y oft)dy' — d z g(3 dd, 
dy = - d x o 7 dd - d y oy dy ' + ( 1 - d x o 7 )dd, 
and we thence have d^=(\ — d x gu)d x — d x qfid y — d x % 7 d z Sec . ; hence 
d= {(1 —d x %cc)d x —d x $fidy—d x % 7 d z \ (X — Y d x o) 
= (1 — d^a)ci — d x pfi . h - d x % 7 .g — d x (V d x y ) 
— a — o{ad/i. -j- hd x (3 J r gd x7 ) 
— (au+hfi+g 7 )d xS 
y ^XiCf V d x Q ; 
and similarly, f'=d,J7d (or d s Y'), that is 
f '=/- f (K a +/^ + C( hv) 
-{b aJ rfft + c 7 )d,p 
— y oY d z p — Y d/],o . 
2G. Completing the reduction, we find 
. / dcu — b — c (SX) 2 
a! —a—% 
Y 
V a 
oXd x p — Yd * §, 
bYdyC-Yd*o, 
, (ecu — a — b (SZ) 2 \ 2. 
c — c s y y a ) — y oZd^ Y dzq, 
-V (Wd 4 + iZd, ? )-Vd,d 4 , 
s’ =S -S ~?( 5Z <A + &X<7 rf )-’ W* 
=h -e (“Y 4 - — V (SX< g +SY d 4 )-YdJ , ? ; 
say these expressions are a'=a-{-Aa, Sec. 
2x2 
