INVESTIGATION OF THE CURVATURE OF SURFACES, 39 
right angles to one another, the line whose direction cosines are proportional to 4, b,, 6, 
must be normal to the quadric. 
This may be seen otherwise. For, equations (7), since 0, may be put in the shape. 

As b. a 
FE One a fe 
“dd db. de 
As similar conditions hold for a,, b,, ¢,, and a, b,, e,, we see thus that the new conditions are 
the analytical expressions for the fact that the axes of the quadric are normals to it. 
To find the directions of the axes of the quadric. 
Applying equations (14) we find that theequations of any axis are 
x = y @ = 
L'ENCRE CELLES 
putting in the several values of & to obtain the three. 



Plane section of a quadric given by the equation in its most general form (the plane passing 
through the origin). 
It will be found that the same formulae of transformation that have been used above 
lead to a simple discussion of the most general case, but as the object of the present paper 
is specially the curvature of surfaces, for which this is not required, it is unnecessary to 
consider it here. 
Axes of a conic in magnitude and direction. 
The properly modified forms of the equations used in investigating the same problems 
for the plane section of a quadric, give simple and direct solutions of these problems. 
CURVATURE OF SURFACES. 
To find the radius of curvature at any point x' y' z' of the section of any surface U = 0 made 
by a plane parallel to a given plane a,x + by + ¢,2+d,= 0. 
First, transform the equations to parallel axes through a’ y' z!, then the equation becomes 
2H2e+ Myt+ N2z)+avr+by¥t+er+2lyz+2mzr+2nxry+de=—0 (23) 
where LZ. M. N, are the first differential coefficients of U, and a, b, c, &c., the second 
differential ccefficients. 
The equation of the plane becomes 
ar+hyteoz—0 (24) 
Secondly, transform next to a new system of co-ordinate planes such that the tangent 
to the section (which is the intersection of (24) with Lx + My + Nz=0) shall be the 
axis of y, and the normal to the section, the axis of z, the plane of the section being the 
plane of y z. 
Now using the properties of determinants, it is easy to put the equations of the tangent 
and of the normal to the section in a simple form. 
