306 
ON CURVATURE AND ORTHOGONAL SURFACES. 
[519 
where coeff. 2p is 
Aa + Hh + Gg (AX + HY+ GZ) 8X 
V V 3 
= yL(hZ- g Y) + hZ-gY^-^(ZSY - YSZ). 
31. And similarly, 
F" = OF + (Ha + B/3 A Fy) d z p + (Ga + F/3+ Cry) d y p 
+ p \(Hd z a + Bd z /3 + Fd z ry) A (Gd y a + Fd y ¡3 + Gd y y)} 
= OF a y. {(HX a BY a FZ) d z p + (GX + FY a CZ) d y p] 
Hg +Bf + Fc 
(HX + BY a FZ) 8Z 
V 
V 3 
Gh + Fb + Gf 
(GX A FY A GZ)8Y] 
V 
V s j 
Gh + Fb + Gf = a>(hY-bX) + hY-bX + aX + dX +hY+gZ, 
Fig A Bf + Fc = — a) (gZ — cX) — gY + cX — wX — ccX — hY — gZ. 
Sum is co{hY-gZ-(b-c)X} + hY-gZ-(b-c)X, which is = coF+ hY-gZ-(B-c)X: 
T’ = 6F + (XSZ-ZSX) (±d zP -&) + (YSX- XSY) (1^ - 
+ ^{ajF+ hY-gZ-(b - c) X}. 
32. We may write 
A" = 6A + 2 (I d,p - p -^j (ZSY -Y8Z) + £ {„A + A), 
B" = SB + 2 (I d,p -(XSZ -ZSX) + £ {coB + B], 
0" =60 +2 (yd,p-?~)(Y8X-XSY)+£{coC +C], 
F" = 6F + (f d,p - ^) (XSZ - ZSX) + (I d,p - ( YSX -XSY) + t (o>F + F\, 
6" = 6G + {fd I p-^f)(YSX-XSY)+{fd 1 p-&)(ZSY-YSZ) + -^[ a G + G}, 
H" =6H+ (f d yP ~^J(ZSY- YSZ ) + (A d x p - (XSZ - ZSX) + £ [coH + H), 
in which equations A, B, &c. are the like functions of d, b, &c. that A, B, &c. 
of a, b, &c.; viz. A — 2hZ—2gY, &c. 
are