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MA THEM A TICS: WILSON AND MOORE 
and the total curvature G is 
G = a-13 — iJL^ = ^s/a = fisA, 
the ratio of the first scalar invariant of ^, or of the determinant 12 of 
the second form to the determinant a of the first form. 
If the surface V2 in 6*5 be projected successively upon the three spaces 
S3 determined by the tangent plane and each of three mutually per- 
pendicular normals issuing from O into the normal space Nz, the three 
projections have the (vector) sum of their mean curvatures and the 
(scalar) sum of their total curvatures equal respectively to the mean 
curvature and total curvature of the given surface at 0. There is there- 
fore a cone, Cone III, of normals such that if any element of the cone be 
chosen as one of the three mutually perpendicular normals, the other two 
may be chosen in such a way (upon Cone II) that the total curvature 
of the three projections are respectively G, 0, and 0. The equation of 
this cone is r . ($3 / — ^) . r = 0, where / is the idemfactor. 
Types of surfaces. — For minimal surface h = 0 , and the indicatrix 
everywhere reduces to an ellipse in some normal plane and with its center 
at the surface-point 0. 
For the surface formed by the tangents to a twisted curve, the in- 
dicatrix reduces to a segment of a line (described twice) reaching from 0 
to the extremity of the vector 2 h. Such a surface is developable, and 
all ruled developables are of this type. 
For any ruled surface the indicatrix lies in a plane passing through 
the surface-point 0, and the ellipse itself passes through 0. The total 
curvature of any real ruled surface (other than developable) is 
negative. 
For a surface of revolution, which is formed by revolving a twisted 
curve parallel to a plane, the indicatrix reduces to a linear segment 
(described twice) centered at the extremity of h. There is a large 
variety of developable surfaces (not ruled) of revolution, of which the 
simplest is perhaps that obtained by revolving the circular helix parallel 
to a plane containing its axis and a Hne perpendicular to the three-space 
in which the helix Hes. 
For a developable (non-ruled), the indicatrix is tangent to three mu- 
tually perpendicular planes through 0, or, if w = 4, to two perpendicular 
lines through 0. A developable is not invariant in type under a pro- 
jective transformation. 
The class of surfaces where the indicatrix at each point lies in a plane 
with 0 is invariant under projective transformations, and these surfaces 
have upon them the characteristic lines of Segre. 
A special type of surface, which have the indicatrix everywhere a 
