248 PROFESSOR CAYLEY ON CURVATURE AND ORTHOGONAL SURFACES. 
41. We have thus arrived at the equation 
((A), . . .X d„ d y , d t )*q = 0 
as the condition to be satisfied by the normal distance g in order that the given surface 
and the vicinal surface may belong to an orthogonal system, viz. this is a partial differ- 
ential 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, . . .JZd s -Yd e , X.d M -Zd„ Yd 2 -Xdfo = Q, 
and 
P Q R 
0P+/1Q+3R, 7iP+iQ+/R, ^P+/Q+c-R 
X Y Z 
if, for shortness, P, Q, It are written to denote Zd y —Yd,, Xd, — Zd x , Y d x — Xd y respec- 
tively, it being understood that in each of these forms the d x , d y , d, operate on the g only. 
Condition that a family of surf aces may belong to an Orthogonal System. 
Article Nos. 42 to 49. 
42. We pass at once to the condition in order that the family of surfaces 
r—r(x, y,z )= 0 
may belong to an orthogonal system, viz., when the vicinal surface belongs to the family, 
we have g proportional to ^ ^ ' w ) ’ anc ^ conc ^ on 
where r is a function of (x, y, z), the first and second differential coefficients of which 
are X, Y, Z, a, b, c,fg , h ; and the equation is thus a partial differential equation of the 
third order satisfied by r. The form is by no means an inconvenient one, but it admits 
of further reduction. 
43. We have d x ^ d y d z y equal to — ^ bX, — oY, —^3 hZ respectively, and 
thence 
dly= — ysfd + Id -f- f + la) + =p(ciX) 2 , 
d y d z y= -yfgh-\-bf Yef +0/ )+ys oYoZ, 
or, as these may be written, 
dly= — yafv— M + « + ^)“Py5(^X) 2 , 
7 7y — ~ yff co +7 +¥) + y.^Y^Z, 
