INVESTIGATION OF THE CURVATURE OF SURFACES. Al 
section on the tangent plane, let us call them cos a, cos f, cos y, respectively. Also let us 
call ¢ the angle between the normal to the quadric, and the normal to the section (whose 


= 5 as bs C3 
direction cosines are —:—»—, whence 
Tae ists 
1 { a; bs Cs 
= — TE EN 
CoN arg es omen Was ares ie 
Hence (25) may be written thus :— 
Radius of curvature of any section 
cos 6 V M?/ + M° + N° (26) 
PT a cos a + b cos ff + ¢ cose y +2 Leos BB cos y + 2 m cos y cos a + 2 n cos a cos fr 

If the section be a normal section cos # = 1. Hence for a normal section the radius of 
curvature 
MN EMI EN 4e ONE 
a cos a + b cos? B + cos y +21 cos B cos y + 2 m cos y cos «a +2 n cos a cos fr 

p= 

From (27) and (26) Mewnier’s theorem is deduced. 
In equation (27) a, B, and y, are connected by the relation 
Lcosa+ M cos fi + Neos y = 0 
Hence. if we take the quadric 
a@+tbytece+2lyztimertincy=—V P+ M+ ND’: (28) 
and observe that any radius vector (r) of the section of it made by the plane 
Lir+My+<Nz—=0 (29) 
is given by the equation 
VITE ME M 
aco a+ b cos B +e cos y +21 cos f5 cos y + 2 m cos y cos w+ 2 n cos a cos f 
2 — 

where 4, f, y, are connected by the relation 
L cos a + M cos f+ Neos y =0 
we see that the square of this radius vector is equal to the radius of curvature of the corresponding 
normal section of the given surface, U = 0. 
Thus for each point of a given surface U= 0, we get a quadric (28) which enables us to find, in 
the manner above described, the radii of curvature of the normal sections through the point in 
magnitude and direction. 
As this quadric, which might be called the “curvature quadric”, differs from that 
which has already been discussed only in having ¥ L*-+ M? + N° on the right hand side 
instead of 1, we may use the results already obtained slightly modified. 
To find the principal radii of curvature (R, and R,) at any point of the quadric U = 0 
This is to find the squares of the semi-axes of the section of the quadric (28) by the 
plane (29). We have therefore first to substitute L, M, N, for a, b, c, in equation (10). Then 
. Sec. IIL, 1882-83. 6. 
