184 
tion, as throwing great light on the subject and leading to 
many results of interest. 
215. If a normal and an oblique section of a surface 
be made by planes passing through the same tangent line 
to the surface, the radius of curvature of the oblique section 
is equal to the projection on its plane, of the radius of 
curvature of the normal section. 
Let the tangent plane to the surface be the plane of 
xy, the point of contact the origin, and the tangent line the 
axis of x; then the normal to the surface will be the axis 
of z ; let OP (fig. 62) be the normal section in the plane of 
zx, OP' the oblique section made by a plane z'Ox through 
Ox, and inclined to the normal section at an Z % 0% = 6; 
NP, NP', ordinates to the two curves corresponding to the 
common abscissa ON=h; also let NP = z, and let h, k, z 
be co-ordinates of P'. Then, (assuming that in any plane 
curve when the axis of the abscissse is a normal at the origin, 
the radius of curvature at the origin is equal to 1 limit of 
ON 2 
R = radius of curvature of OP at O = i limit of , 
^ NP ’ 
R'= radius of curvature of OF' at 0 = | limit of 
2 NP > ’ 
r 
% 
= sec 9. limit of —. 
z 
z' sec 0 
— = limit of ^ — = limit of 
But since the plane of xy is the tangent plane at 0, 
p = 0, q = C, and z'= h 2 + shk + \tk 2 + &c. 
and making k = 0, 
% = 1 rh 2 + • h 2 + &c.; 
dx 
k ' 2 
r + 2s- + t + &c, 
h 
6. limit of - "" ’ —■— sec 9, 
, dr 
r + 4 -—. h + & c. 
a dx