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ON THE SURFACES WITH PLANE OR 
[886 
different planes) the two isosceles triangles NPP', OPP' on a common base PP', then 
the angle OPN is equal to the angle OP'N. For take P, P' consecutive points on 
a spherical curve of curvature; then at P, P' the normals of the surface meet in a 
point N, and the normals (or radii) of the sphere meet in the centre 0, and we have 
angle OPN = angle OPN, that is, at each of these points the inclination of the normal 
of the surface to the normal of the sphere has the same value; and this value being 
thus the same for any two consecutive points, must be the same for all points of the 
curve of curvature. The proof applies to the plane curve of curvature; but in this 
case, the fundamental theorem may be taken to be, a line at right angles to the base 
PP' of the isosceles triangle NPP' is equally inclined to the two equal sides 
NP, NP'. 
A surface may have one set of its curves of curvature plane or spherical. To 
include the two cases in a common formula, the equation may be written 
k (x 2 + y 2 + z 2 ) — 2ax — 2by — 2cz — 2u = 0; 
k = 1 in the case of a sphere, = 0 in that of a plane; and the expression a sphere 
may be understood to include a plane. I write in general A, B, 0 to denote the 
cosines of the inclinations of the normal of the surface at the point (x, y, z) to the 
axes of coordinates (consequently A 2 + B 2 + G 2 = 1). Hence considering a surface, and 
writing down the equations 
k (,x 2 + y~ + z 2 ) — 2ax — 2by — 2cz — 2u = 0, 
(kx — a) A + (ky — b)B + (kz — c)G = l, 
where (a, b, c, u, l) are regarded as functions of a parameter t. The first of these 
equations is that of a variable sphere; and the second equation expresses that at 
a point of intersection of the surface with the sphere, the inclination of the tangent 
plane of the surface to the tangent plane of the sphere has a constant value l, viz. 
this is a value depending only on the parameter t, and therefore constant for all points 
of the curve of intersection of the sphere and surface: by what precedes, the curve 
of intersection is a curve of curvature of the surface, and the surface will thus have 
a set of spherical curves of curvature. 
Supposing the surface defined by means of expressions of its coordinates (x, y, z) 
as functions of two variable parameters, we may for one of these take the parameter t 
which enters into the equation of the sphere ; and if the other parameter be called 6, 
then the expressions of the coordinates are of the form x, y, z = x(t, 6), y(t, 6), z (t, 6) 
respectively; these give equations dx, dy, dz = adt+ a'dd, bdt + b'd6, cdt+c'd0, where of 
course (a, b, c, a', V, c') are in general functions of t, 0; and we have A, B, 0 pro 
portional to be — b'c, ca’ — c'ci, ab' — a'b, viz. the values are equal to these expressions 
each divided by the square root of the sum of their squares. In order that the 
surface may have a set of spherical curves of curvature, the above three equations 
must be satisfied identically by means of the values of 
a, b, c, u, l, A, B, G, x, y, z,