40 À A. JOHNSON ON SYMMETRICAL 
The equations of the tangent to the section will be 
where &, b,, ¢, are the determinants obtained from 
| | a, bi, Cy 
zara a 
by removing each of the columns in turn. For, we shall thus have 
H&+bb +e%=0, La +Mb+Ne—0 
expressing the fact that the tangent to the section lies in the tangent plane to the surface 
as well as in the plane of the section. Again, the equations of the normal to the section, 
dde : x y Zoe 
which is perpendicular to the tangent and to — = — = — will be 
ay 1 Ci 
V4 Le y as (4 
TRE: G 
in the same manner as before. 
Hence the new co-ordinate planes will be 
Hrtbhytaz=0 oX¥—0 
e+ bytoz=0 or Y=0 
a,c+bytez=0 orZ=—0 
The formulae of transformation will therefore be the same as before (4). 
If we effect the transformation and then make X = 0, we shall get the equation of the 
section, which, as the axis of y isa tangent to the curve, and of z, a normal, will be of 
the form (using small letters instead of large) 
@Gzteyt2nyz4+Cut+oy-+ete.=—0 
| : : Ô 5 
The radius of curvature of this will be 3e Hence we have only to seek the ccefficients 
of = and y” in the transformed equation. 
These will be found to be 

0 — 9 La,+ Mb,+ Ne; 

T3 
pa fo (Go bs Cr) 
Ty 
using same notation as before. 
Hence the radius of curvature is equal to 
as 8 C3 
Tj Le Ae CNRS 
i T3 Ts Ta 25 
3 l 
= lea +0 bE + 0 ef +21 4 -+2m ea, +2n a,b 
My D © : : . c 30 
PRE 0 the direction cosines with respect to the original axes Of the trace of the 
T2 Te 

Since 
2 
