244 PROFESSOR CAYLEY ON CURVATURE AND ORTHOGONAL SURFACES. 
33. Similarly, the value of flF is 
*F= Y ( — * (/ m+h) + SX5Y - i (SY d 4 + HXd d )-Vd,d ,, f ) 
~z(-f (?»+£)+$ SZ5X -i (SZ d, s +iXd 4 )-Vd,d,^ 
W 8 , , r > S iYa fSZ" 
— — y 1(6 — <?)*; + & — c\ -j-“y3 yf 
- 1 iYd, s +~ iZd 4 - V<% +V<e s ) 
— V(M, — gd z —{b— c)cl x )o, 
which is 
= | (_Fa,-P)+^l (XSZ-Z£X) + s |l (YSX-XSY) 
+ j-i(YSY-ZSZ)+V(5-c)j<t ? 
+ {-|sX+~8Y -VA }<J, e 
+ ( fsX+^SY +Vg }^ ? 
+ (-VY(7/Z ( +VZrfX-+VX^-VX«)§. 
Hence F" is equal to the foregoing expression, together with the following terms : 
+ *(RH-F)- 8 ^ (XSZ-ZSX)-Sg (YSX-XSY) 
+| (YSX-XSY)^ + ^(XSZ-ZSX)cl 4 , 
which destroy certain of the foregoing terms ; viz. we thus have 
F'=(-i(Y 8 Y-ZSZ)+V(J-c)} < ? 4 +j| 8 Y-VA}i, e + {-^Z-vd<?, ? 
+ V( — Y d x d y -j- Z d z d x -f- Xr7| — X<S)g . 
34. We thus have 
A"= . 2(v« ? -~)<?,g-2(v/ J -™)^+2V(-Z<J/, +Y<?A) ?> 
B" = -2(vf-^P)d 4 . +2(vh-™y, ? +2Y(-XdJ,+ZdJ,) ? , 
c' = +2(y/-^)4 ? _2(v< ? -Y?)^ . +2V(-Y dA+xa-Ah 
