305 
519] ON CURVATURE AND ORTHOGONAL SURFACES, 
whence term in { } is 
Af, 
A77, 
Af j + 
aAf + AA?; + $rAf, 
/¿Af 4 ¿At? + /Af, 
¿/Af +/A?7 + cAf 
*f, 
, 
«f ! 
A 
F 
Z 
A, 
F , 
Z 
f 
K 
which might be written 
Af, 
Atj, 
Af 
- 
«Af, 
SA?;, 
8Af 
«f, 
Brj, 
Sf 
f , 
V > 
K 
A, 
F , 
A , 
Y , 
z 
but it is perhaps more convenient to retain the second term in its original form. 
29. As regards the first line, we have 
A' = 2h'Z' - 2g'Y' 
= 2 (h + Ah) (Z- Vd z p) -2 (g + Ag) (F — Vd y p) 
= A + 2 (ZAli — YAg) — 2V(hd z p - gd y p), 
with similar expressions for the other coefficients. Attending only to the terms of the 
first order, we thus obtain 
A' = A + 2 (Z Ah — YAg) — 2 V (Jid z — gd y ) p, 
B' =B + 2 (XAf— ZAh) - 2 V(fd x - hd z ) p, 
C’ = C + 2 (YAg - XAf)- 2 V (gd y - fd x ) p, 
F' = F + YAh — Z Ag — X{Ab — Ac) —V(Jid y — gd z ~{b — c)d x ) p, 
G' = G + Z Af — XAh — Y (Ac — Aa) — V (fd z — hd x — (c — a) d y ~) p, 
H' = H + XAg - YAf -Z(Aa- Ab) - V(<gd x -fd y ~{a~b)d z ) p, 
say these are A' = A + 6A, &c., where 6 is a functional symbol; we thus have 
V, = n, + n, f)> + 2(4, At,, Af), 
which, for shortness, I represent by 
,, v, f) s ; 
and I proceed to complete the calculation of the coefficients A", B", &c. 
30. We have 
A" = 6A + coeff. f 2 in 
2 [(Af + Eg + Gf) Af + (iff + B n + Af) A v + (<7f + F v + (7f) Af] 
= 6A + 2 (Ad x pa + Hd x p/3 + Gd x p<y), 
that is, 
A" = 6A + 2 y(AX + HY+ GZ)d x p 
+ 2p (Ad x ct + Hd x fi + Gd x ry), 
C. VIII. 
39