519] ON CURVATURE AND ORTHOGONAL SURFACES. 303 
and we thence have d* = (1 - d x pa) d x - d x p/3d y - d x pyd z &c.; hence 
Oj {(1 d x pa) d x d x p(3d y d x pyd z } (X — Vd x p) 
= (1 - d x px) a - d x pfi. h - d x py .g-d x ( Vd x p) 
= a- p (ad x a + hd x /3 + gd x y) 
- (ax -f h/3 + gy) d x p 
1^ 3 .V d/p p Fd x *p , 
and similarly, /' = (or d z >Y'), that is 
/' =/- p (gd y a +fd y fi + cdyj) 
~ ( f J a +/£ + oy) d y p 
8~Y^d z p Vd y d z p. 
26. Completing the reduction, we find 
a’ = a — p 
(aw — b — c (BX) 2 \ 
— y 8Xd x p - Vd x -p, 
V v 
V s J 
V = b-p 
(bw—o — a (SF) 2n i 
\ -ySYdyp-Vdy’p, 
V V 
F 3 ) 
(cw—a — b (8Z) 2 ' 
) - y 8Zd z p - Vd z 2 p, 
V V 
F 3 j 
Q_ 
1 
X 
II 
(wf + f 8Y8Z \ 
V V V s ) 
Y^Yd z p + 8Zd x p ) - 
9' =9~P 
(wg + g 8Z8X \ 
\V W~ ) ~ 
Y (8Zd x p + 8Xd y p) — 
ti =h-p 
(wh 4- h 8X 8 F\ 
V v V s ) 
^ (8Xd y p + 8Yd z p) — 
say these expressions are a' — a + Aa, &c. 
27. Taking £, g, £ for the coordinates, referred to P as origin, of a point on the 
given surface near to P, and £', V, £' f° r the coordinates, referred to P' as origin, of 
the corresponding point on the vicinal surface, the relation between g, £' and £, £ 
is the same as that between dx, dy’, dz and dx, dy, dz; viz. we have 
iij — (1 — d x pa) £ dypx . g d z px . , 
g-- d x pl3. £ +(l-d y pp)g -d z p@ . £', 
K = — dxpy ■ f - dypy .g + (1 - d z py) £';