314 
ON CURVATURE AND ORTHOGONAL SURFACES. 
47. It remains to prove the above-mentioned identities. 
To reduce the term (A,. Q8X, 8Y, 8Z) 2 , we have 
A8X + H8Y+G8Z 
that is, 
and similarly 
whence 
= A(aX + hY+gZ) + H(hX+bY+fZ) + G(gX+fY + cZ) 
= X{ co (JiZ — gY) + JiZ-gY} 
+ Y {-co (f Y - bZ) -(fY- bZ) - coZ-lZ] 
+ Z{ co (fZ —cY)+ fZ-cY +C0Y+8Z} 
= « (Z8Y- Y8Z) + (Z8Y— Y8Z) + (Z8Y- Y8Z), 
A8X + 7TSF+ G8Z=co (Z8Y - Y8Z) + 2 (Z8Y - Y8Z)- 
H8X + B8Y +F8Z = co (X8Z - Z8X) + 2 (X8Z - Z8X), 
G8X +F8Y + C8Z = co (Y8X - X8Y) +2 (Y8X - X8Y), 
(A,..J8X, 8 Y, 8Zy = - 2 
8X, 
8Y, 
8Z 
x, 
Y, 
Z 
8X, 
SY, 
8Z 
48. Now, from the equations AX + HY + GZ = Z8Y — Y8Z, &c. we have 
value of twice the foregoing determinant 
2 det. = 2 {(aX + hY + gZ) (AX + HY + GZ) 
+ (hX + bY+fZ) (.EX + BY + FZ) 
+ (gX + fY + cZ)(GX + FY + GZ)}; 
and subtracting herefrom the function ((J.), .. ..), which is 
(BZ 2 +GY 2 — 2FYZ)a 
+ (GX 2 + AZ 2 - 2GZX) b 
+ (AY 2 + BX 2 — 2HYZ)c 
+ 2 (-AYZ-FX 2 +GXY+HXZ)f 
+ 2 (- BZX 4- FX Y - GY 2 +HYZ)g 
+ 2(-GXY+FXZ+ GYZ-HZ 2 )h, 
the difference is found to be 
= a{(A,..\X, Y, Z) 2 + A V 2 } 
+ b {(A,..IX, Y, Zf + BV 2 } 
+ c{(A,..lX, Y, Z) 2 + GV 2 } 
+ 2/ {(A + B + G) YZ +FV 2 } 
+ 2g{(A+B + G)ZX + GV 2 } 
- 2 h \(A + B+C)XY + HV%