MA THEM A TICS: WILSON AND MOORE 
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^= —{dy -yCj-dM, where the dot denotes the inner product,Mn com- 
plete analogy with the expression— Jy ^/n, in terms of the unit normal n, 
in ordinary surface theory. Moreover, the square of dM is (fi^M)2 = 
— Gds"^ + 2h • ^Jr, and thus gives what may be called the third fundamental 
form. (In the three-dimensional case, this is associated with the 
spherical representation.) 
The indicatrix. — If b = i (a — iS) and if accents be used to denote 
quantities corresponding to a direction at an angle B to X, we find 
ix' = IX cos 26 - b sin 26, d' = 8 cos 26 jx sin 26. 
This means that when we turn about a point of the surface the vectors 
m', 8' describe an ellipse (with center at the extremity of the mean 
curvature h) of which any two positions of m', 6' are conjugate radii, 
the eccentric angle between ix or 5', b being twice the angle 6. This 
ellipse we call the indicatrix."* The vectors a , jS' originating at the sur- 
face-point 0 and terminating in the indicatrix describe a cone of normals, 
Cone I. 
The indicatrix is determinative of a large number of properties con- 
nected with the curvature of a surface at a point. As the surface is 
two-dimensional and the indicatrix with the surface-point 0 determines 
a normal three-dimensional space, the properties of curvature are as 
general for a surface V2 in a five-dimensional space 6*5 as in 5n, and the 
further developments may be given in the assumption that n = S. 
Consecutive normal plane three-spaces Nz intersect in a line. These 
normal lines all pass through a point O' lying upon the perpendicular 
from 0, to the plane of the indicatrix and generate a quadric cone, 
Cone II. The relation between Cones I and II is reciprocal in the 
sense that each element of either is perpendicular to some tangent plane 
of the other. Hence if <^ be the linear vector function which occurs in 
the equation r.<i>.r = 0 of Cone II, referred to its vertex, the equation 
of Cone I, referred to its vertex, is r-$~^T = 0. 
The function $ may be written as $ = (hh — jxix — bb)/a, where 
a = \ars \, in terms of the mean curvature h and a pair of conjugate radii 
of the indicatrix. $ is the self conjugate part of the function 
n = 
= Yn Y22 — yi2 y2i 
yn yi2 
y2i y22 
formed as the determinant (with vector elements) of the second form ^. 
We have 
^ = I yn 722 + i y22 yu - yi2 721, 
