790] 
GEOMETRY. 
581 
The line through the point at right angles to the tangent plane is called the 
normal; its equations are 
g-X 7) - y _ g-z 
df ~ df df' 
dx dy dz 
In the series of tangent lines there are in general two (real or imaginary) lines, 
each of which meets the surface in a second consecutive point, or say it has three- 
point intersection with the surface; these are called the chief-tangents (Haupt- 
tangenten). The tangent-plane cuts the surface in a curve, having at the point of 
contact a node (double point), the tangents to the two branches being the chief-tangents. 
In the case of a quadric surface the curve of intersection, qua curve of the 
second order, can only have a node by breaking up into a pair of lines; that is, 
every tangent-plane meets the surface in a pair of lines, or we have on the surface 
two singly infinite systems of lines; these are real for the hyperbolic paraboloid and 
the hyperboloid of one sheet, imaginary in other cases. 
At each point of a surface the chief-tangents determine two directions; and passing 
along one of them to a consecutive point, and thence (without abrupt change of 
direction) along the new chief-tangent to a consecutive point, and so on, we have on 
the surface a chief-tangent curve; and there are, it is clear, two singly infinite series 
of such curves. In the case of a quadric surface, the curves are the right lines on the 
surface. 
40. If at the point we draw in the tangent-plane two lines bisecting the angles 
between the chief-tangents, these lines (which are at right angles to each other) are 
called the principal tangents*. We have thus at each point of the surface a set of 
rectangular axes, the normal and the two principal tangents. 
Proceeding from the point along a principal tangent to a consecutive point on 
the surface, and thence (without abrupt change of direction) along the new principal 
tangent to a consecutive point, and so on, we have on the surface a curve of curvature; 
there are, it is clear, two singly infinite series of such curves, cutting each other at 
right angles at each point of the surface. 
Passing from the given point in an arbitrary direction to a consecutive point on 
the surface, the normal at the given point is not intersected by the normal at the 
consecutive point; but passing to the consecutive point along a curve of curvature 
(or, what is the same thing, along a principal tangent) the normal at the given point 
is intersected by the normal at the consecutive point; we have thus on the normal 
two centres of curvature, and the distances of these from the point on the surface are 
the two principal radii of curvature of the surface at that point; these are also the 
radii of curvature of the sections of the surface by planes through the normal and the 
two principal tangents respectively; or say they are the radii of curvature of the 
* The point on the surface may be such that the directions of the principal tangents become arbitrary ; 
the point is then an umbilicus. It is in the text assumed that the point on the surface is not an 
umbilicus..