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