519] 
ON CURVATURE AND ORTHOGONAL SURFACES. 
311 
40. The reduction depends on the following auxiliary formulae : 
a(a)+h(h)+g(g)= V8V-X8X, 
h „ + b „ +f „ = - Y8X, 
9 „ + />» +c „ = — Z8X, 
a(h) +h(b) + g (/) = - X8Y, 
h „ +b „ +f „ = V8V-Y8Y, 
9 „ +/„ + c ,, = -Z8Y, 
a (y)+Hf)+g(c)= —X8Z, 
h „ +b „ +/„= — Y8Z, 
9 +f » +c „ = V8V— Z8Z, 
where, for shortness, I have written 8X, 8Y, 8Z to stand for aX +kY+gZ, hX + hY+fZ, 
gX+fY+cZ respectively, and V8V for X8X + Y8Y+Z8Z, (= a,.JX, Y, Zf. 
From these we immediately have 
(a)8X + (h) 8Y + (g) 8Z = V(X8V- V8X), 
(h) 8X + (6) 8Y + (f)8Z=V{Y8V-V8Y), 
(g)8X + (f)8Y+{c) 8Z=V(Z8V -V8Z). 
Hence, in the coefficient of d x p, the first line is 
= 2V(— Y8Z + Z8Y), 
and the second line is 
= y{VZ(Y8V- V8Y) - VY(Z8V-V8Z)J, = 2V(Y8Z - Z8Y) ; 
so that the sum, or whole coefficient of d x p, is = 0. Similarly, the coefficients of d y p 
and d z p are each = 0. 
41. We have thus arrived at the equation 
((-d.)j • • dy, d z )-p=0 
as the condition to be satisfied by the normal distance p in order that the given 
surface and the vicinal surface may belong to an orthogonal system, viz. this is a 
partial differential equation of the second order, its coefficients being given functions 
of X, Y, Z, a, b, c, f, g, h, the first and second differential coefficients of r (where 
r = r(x, y, z) is the equation of the given surface). 
The equation, it is clear, may also be written in the two forms 
(A,...\Zd v — Yd z , Xd z - Zd x , Yd x — Xd y ) 2 p — 0, 
and 
P Q , P 
aP + hQ + gR, hP + bQ + fR, gP + fQ+ cR 
X Y , Z 
p= 0, 
if, for shortness, P, Q, R are written to denote Zd y —Yd z , Xd z —Zd x , Yd x —Xd y 
respectively, it being understood that in each of these forms the d x , d y> d z operate on 
the p only.