519] ON CURVATURE AND ORTHOGONAL SURFACES. 299 
These values give 
AX + HY+ GZ = Z8Y — YhZ, 
HX + BY + FZ = X8Z - Z8X, 
GX + FY + CZ = Y8X-X8Y; 
whence also 
(A,...IX, Y, Z) 2 = 0, 
as is, in fact, at once obvious from the determinant-form; and also 
A+B+C=0. 
17. Writing for shortness 
(a, b, c, /, g, h) = (be -/ 2 , ca - g 2 , ab - h~, gh - af, hf- bg, fg - ch), 
we find 
Aa + Hh + Gg = (o (hZ — gY) + liZ — gY, 
Hh + Bb + Ff = co (fX - hZ ) +fX - hZ, 
Gg+Ff+Gc = to (gY — fX ) + gY—fX; 
whence 
(A, ...$«, ...) = 0. 
18. By what precedes, we have 
((4), n, f)» = 0 
for the equation of the two principal planes, where the coefficients (A), (B), &c. are 
functions of A, B, &c. and of X, Y, Z, as mentioned above. These coefficients satisfy 
of course the several relations similar to those satisfied by (a), (b), &c., and other 
relations dependent on the expressions of A, B, &c. in terms of a, b, &c. and X, Y, Z. 
19. Proceeding to consider the coefficients (A), (B), &c., we have then 
(A) + (B) + (G) — (A + B+G)V 2 — (A,. .$X, Y, Z)\ 
that is 
(A)+(B) + (G) = 0. 
Observing the relation A + B + G = 0, the equations analogous to 
(a) --= (b + c) V 2 - (a + b + c) X 2 + &c., are (A) = - A V 2 + X8'X -Y8'Y- Z8 Z, &c. 
if for a moment we write 8'X, 8'Y, 8'Z to denote the functions 
AX + HY+GZ, HX + BY+FZ, GX + FY+CZ. 
But, from the above values, X8'X + Y8'Y+ Z8'Z = 0, or the equation is (A) = — A V- + 2A 8 A, 
that is = — A V 2 + 2A (Z8Y- Y8Z). The equation for (F) is (F) = - FV 2 + Y8’Z + Z8'Y, 
where Y8'Z + Z8'Y is = Y(Y8X - X8Y) + Z(X8Z- Z8X), viz. this is 
= (Y 2 — Z 2 ) 8X - XY8Y+ XZ8Z. 
38—2