250 PROFESSOR CAYLEY ON CURVATURE AND ORTHOGONAL SURFACES. 
or, written at full length, it is 
(A )la + (B)U + (C)Sc + 2(F)&/+ 2(G)ty + 2(11 )lh 
— 2{(A )a +(BjZ +(C ))c +2(F)/+2(G)^ + 2(H)A} = 0, 
where the coefficients are given functions of X, Y, Z, a, b , c,f, g, h, the first and second 
differential coefficients of r; and S is written to denote Xd x Y d y 4- ZY. 
47. It remains to prove the above-mentioned identities. 
To reduce the term (A, . .($AX, $Y, bZf, we have 
m+mY+Gffi 
= A(«X -I- hY+gZ)+ H(AX + 5Y+/Z) + G(gX +/ Y + cZ) 
= X{ co(h Z -^Y)+AZ -7jY] 
+Y{-*>(fY-bZ)-(jY-bZ)-uZ-bZ} 
-1-Z{ a>(fZ—cY)+JZ—cY +TY+TZ} 
= *(ZSY- Y^) + (ZlY~YlZ) + (ZlY- YlZ), 
that is 
whence 
A&X+H&Y+GaZ=fi;(Z&Y — Y&Z ) + 2(Z5Y -Y5Z ), and similarly 
H&X+BSY + FcSZ =«(XSZ - za ) + 2(X§Z - ZlX ), 
GciX + Fc>Y + C^Z = co{ Y&X - XYY) -f 2( Y§X - XS Y), 
(A, . .X^X, &Y, ciZ) 2 = — 2 
ax, 
&Y, 
£Z 
x, 
Y, 
z 
LX, 
8Y, 
iz 
48. Now, from the equations AX+HY + GZ = ZoY— -Yc>Z, &c., we have for the value 
of twice the foregoing determinant 
2 det . = 2 { (aX +TtY-\-gZ) (AX + FI Y + GZ ) 
+ (AX-f FY + /Z)(HX+ BY + FZ ) 
-f^X+J'Y+FZ )(GX +FY -J-CZ)} ; 
and subtracting herefrom the function ((A), . . .\a, . .), which is 
= (BZ 2 + CY 2 — 2FYZ ja 
-f(CX 2 +AZ 2 -2GZ X)b 
+ (AY 2 + BX 2 - 2F YZ )F 
+ 2( - AYZ - FX 2 + GXY + HXZ )J 
+ 2(-BZX-FFXY-GY 2 +HYZ)g 
-F2( — CXY+FXZ+GYZ —HZ 2 )h, 
