PROFESSOR CAYLEY ON CURVATURE AND ORTHOGONAL SURFACES. 
241 
but 
Ag 
+ H/7 + G£ = 
=Z^ 
-Y^+ 
a , 
A, 
9 
x, 
Y, 
Z 
?? , 
K 
Hg 
+B fl +Ff= 
=xa£- 
-Z&I + 
A , 
f 
X, 
Y, 
z 
v 
§ 9 
K 
Gg 
+ Fp,+C£ = 
II 
cJ V? 
9 > 
/» 
c 
X, 
Y, 
Z 
1 , 
*7 s 
K 
5 
whence tei 
•m in 
{ } is 
A§, 
A y, 
A? 
+ 
aAg + hi Ay -j-gA£, 
AA£ + AAp?+/A£, 
tfAg+/Xj? + cA£ 
, 
h , 
% 
X 
5 
Y 
Z 
X , 
Y , 
Z 
5 
5 
, 
S 
which mig 
ht be 
written 
A£, 
Art, 
A? 
— 
bA%, 
Sr 
<3 
<1 
^ , 
by , 
1 » 
>7 5 
X , 
Y, 
Z 
X , 
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(7* + Ah)(Z — VcL§) -2(g-\- Ag) ( Y — Vc^f) 
= A + 2(ZAh-YAg) - 2 Y^-gd^), 
with similar expressions for the other coefficients. Attending only to the terms of the 
first order, we thus obtain 
A'= A+2(Z Ah- Y Ag)-2V( hd z -gd y ) § , 
If — P, -j- 2(X A/- Z Ah)- 2Y (fd x - hd z )g, 
C'= C +2(Y A g - XAf) - 2V( gd.-fd.) )g, 
F =F +YAA-Z Ag-X(Ah-Ac)-V(hd y -gd z -(b-c)dJo, 
G' = G + Z A/- X A h - Y ( A c- A a) - Y{fd z -M x -(c- a)d v ) Si 
H' = H + X A g- YA f— Z{Aa— Ah) - V( gd x -fd y -{a-b )d z ) s , 
