582 
GEOMETRY. 
[790 
normal sections through the two principal tangents respectively. Take at the point 
the axis of z in the direction of the normal, and those of x and y in the directions 
of the principal tangents respectively, then, if the radii of curvature be a, b (the signs 
being such that the coordinates of the two centres of curvature are z = a and z = b 
respectively), the surface has in the neighbourhood of the point the form of the 
paraboloid 
X 2 y 2 
2 = 2a + 2b ’ 
and the chief-tangents are determined by the equation 0 = + |rr • The two centres 
of curvature may be on the same side of the point or on opposite sides; in the 
former case a and b have the same sign, the paraboloid is elliptic, and the chief- 
tangents are imaginary; in the latter case a and b have opposite signs, the para 
boloid is hyperbolic, and the chief-tangents are real. 
The normal sections of the surface and the paraboloid by the same plane have 
the same radius of curvature; and it thence readily follows that the radius of curvature 
of a normal section of the surface by a plane inclined at an angle 6 to that of zx 
is given by the equation 
1 _ cos 2 6 sin 2 6 
pa b 
The section in question is that by a plane through the normal and a line in 
the tangent plane inclined at an angle 6 to the principal tangent along the axis 
of x. To complete the theory, consider the section by a plane having the same trace 
upon the tangent plane, but inclined to the normal at an angle cf>; then it is shown 
without difficulty (Meunier’s theorem) that the radius of curvature of this inclined 
section of the surface is = p cos cp.