519] ON CURVATURE AND ORTHOGONAL SURFACES. 307 
The value of 6A is 
BA = tZ(j- ¿(A., + h) + £ SXSf} -1 (STd.p + SXd y p)- Vd x d y pj 
— 2 Y ( | — y(9 0} +9) + y, SZSxj — y (SZd % p + SXd z p) — Vd x d z p^ 
— 2V(hd z — gdy) p, 
which is 
+ |?,8Z (Z8Y — Y8Z) 
— 2 V (hd z — gdy) p 
- y (Z8Y- Y8Z) d xP -2V(Zdy - Yd z )d x p. 
Hence the value of A" is equal to the last-mentioned expression, together with 
the following terms:— 
+ ^(o>A +A)-1? SX (Z8 Y-nZ) + y(Z8Y- YSZ) d x p, 
which destroy certain of the foregoing ones; viz. we have 
A" = (2 Vg - ) d,p - 2 (FA- d z p — 2V(Zd„ - Yd z ) d„p. 
33. Similarly, the value of 6F is 
6F = F ( - £ (Ä® + Ä) + 3ZSF- y (8Yd x p + 8Xd y p) - F<4<4/>) 
- Z y (^ft) + ¿7) + 8Z8X - y (8Zd x p + 8Xd z p) - Vd x d z p^j 
-X(-^{(b-c)o> + b-c} + p J^- p ^-y8Yd y p + ^8Zd z p-Vd y °p+Vd>p) 
- V (hd y - gd z -(b-c) d x ) p, 
which is 
= F(-Fco-F) + p ^(X8Z-Z8X)+^(Y8X - X8Y) 
+ j— y (Y8Y-Z8Z) + V(b - c)j d xP 
+ j-^SX + ^SF - Vh j dy P 
+ | y8X + ^8Y +Vg j d z p 
+ (- VYd x d y + VZd x d z + VXdy 2 - VXdi) p. 
= -y(a>A +A) 
-^{Zd y -Yd z )p 
39—2