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, 
?