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) £';