230 PROFESSOR CAYLEY OX CURVATURE AND ORTHOGONAL SURFACES. 
where the coefficients depend on the first and second differential coefficients of U, if 
U=0 is the equation of the given surface. Now, considering the given surface as 
belonging to a family, or writing its equation in the form r — r(x, y , s) = 0 (the last r 
a functional symbol), the condition in order that the vicinal surface shall belong to this 
family, or say that it shall coincide with the surface r(x, y , s) = 0, is where 
Y=\/X 2 q- Y 2 -f Z\ if X, Y, Z are the first differential coefficients of r(x, y, z ), that 
is, of the parameter r considered as a function of the coordinates ; we have thus the 
equation 
((A), (B), (C), (F), (G), (H)I<L d„ <1)4=0, 
viz. the coefficients being functions of the first and second differential coefficients of r, 
and V being a function of the first differential coefficients of r, this is in fact a relation 
involving the first, second, and third differential coefficients of r, or it is the partial 
differential equation to be satisfied by the parameter r considered as a function of the 
coordinates. After all reductions, this equation assumes the form previously mentioned. 
On the Curvature of Surfaces. Article Nos. 1 to 21. 
1. Curvature is a metrical theory having reference to the circle at infinity ; each point 
in space may be regarded as the vertex of a cone passing through this circle, say the 
circular cone ; a line and plane through the vertex are at right angles to each other 
when they are polar line and polar plane in regard to the cone ; and so two lines or two 
planes are at right angles when they are harmonics in regard to the cone, that is, when 
each line lies in the polar plane, or each plane passes through the polar line of the 
other. A plane through the vertex meets the cone in two lines, which are the “ circular 
lines” in the plane and through the point; a line through the vertex has through it 
two tangent planes, which might be called the “circular planes” of the point and 
through the line ; but the term is hardly required. Lines in the plane and through the 
point, at right angles to each other, are also harmonics (polar lines) in regard to the 
two circular lines. 
2. Consider now a surface, and any point thereof; we have at this point a tangent 
plane and a normal. The tangent plane meets the surface in a curve having at the 
point a node, and the tangents to the two branches of the curve (being of course lines 
in the tangent plane) are the “ chief tangents” of the surface at the point. 
3. The chief tangents are the intersections of the tangent plane by a quadric cone, 
which may be called the chief cone ; but it is important to observe that this cone is not 
independent of the particular form under which the equation of the surface is presented. 
To explain this, suppose that the rational equation of the surface is U=0 ; taking £, y, £ 
as current coordinates measured from the point as origin, the equation of the chief cone 
is (|<L + ^fi~£cb) 2 U : =0, where x, y , z denote the coordinates of the point. But it is in 
the sequel necessary to present the equation of the surface in a different manner ; say 
