304
ON CURVATURE AND ORTHOGONAL SURFACES.
[519
or conversely
£' = (1 + d x pa) £ 4 d y pa . 77 + d z pu . £
77' = . f 4 (1 4 <4/3/3) t; 4 d z pfi. £
K' = d x py . £ + d y py . 77 + (1 + d z py) £
say £’> Vi — I 4 Af, ^ + A77, £ + A£; hence
Z'f 4 FV + ¿T = (* " Fi4/>) (I + A© 4 &c.
= Z|+Ft7 + ^
+ ZA£+FAt7 + ZA£
- v (%d x p 4 rjdyp 4 £d z p),
where second line is
(Za 4 Yß 4 Zy) (Çd x p 4 Tidyp 4- Çd z p)
4 p {(Xd x a 4 Yd x ß 4 Zd x y) !- 4 (Xd y a 4 Yd y ß 4 Zd y y) 77 4 (Xd z a 4 Yd z ß 4 Zd z y) Ç).
But
Xd x « + Yd,ß + Zd,7 = - [J 8Z = 0,
Zd ?/ a 4 Ydyß 4 Zd y y = 0,
Xd z a 4 Yd z ß 4 Zd z y = 0,
or second line is = V(%d x p 4 77d y p 4 £d z p) ; and we have therefore
Z'f 4 FV 4 Z'f = Zf 4 F77 4 Z£.
5', C", F' f G', tf'$f, V, O 2 ;
viz., to the first order in p, this is
We require
= (A', ...$£ 1?, (ГУ
4 2(4,...M A77, Ag$f, 77, 0-
28. Here second line is
2 {(Л f 4 Я77 4 Сф A£ 4 (#£ 4 B77 4 FÇ) Д77 4 (Gf 4 Z77 4 CÇ) Af :
Лf 4 Д77 4 G£ = ZS77 - Y8Ç +
a ,
Ä ,
9
X,
F,
Z
b
V >
Г
H% + B v 4Z£ = ZS£-ZS£4
Ä,
b ,
/
Z,
Y,
z
1,
V >
?
Gf 4 Z77 4 OÇ = FBf 4 Z877 4
9 >
Z,
с
X
Y,
Z
v,
?