332 WILSOX AND MOORE. 



G will reduce to a single term. In other words: It is 'possible in w^ 

 ways so to select three perpendicular normals Zi that one of the projec- 

 tions has the entire total curvature of the surface and the other two have 

 zero total curvatures. 



If now the dyadics $, $"\ 'J>sl — $ = H, be referred to their prin- 

 cipal directions, 



a2 fc2 ^2 



$ = fl'^ii + //jj - c-kk, (97) 



E= (b-- c')ii + (a2 - c')n + (a^ + 62)kk, 



where i, j, k, stand for Zi, Zo, Z3. If c^ is less than a~ and 6^ the cone 

 defined by S is not real and the above resolution of the surface is 

 impossible in the real domain, — as must be expected when Cone II 

 is so narrow as to have no vertical angles as great as 90°. In case 

 G = 0,^s = and Cones II and III coincide. Hence: The condition 

 6' = implies that Cone II is a cone circumscribed about a trirectangular 

 trihedral angle. 



As Cone I is reciprocal to Cone II, Cone I in that case must ])e 

 inscriptable in a trirectangular trihedral angle or: when G = 0, the 

 indicatrix must be tangent to three viutxially perpendicular planes through 

 the surface-point. In the special case wdiere the indicatrix lies in a 

 plane through the surface point the condition requires that the conic 

 subtend an angle of 90° at the surface point or that the surface point 

 must lie upon a circle of radius (a^ + 6')* concentric with the 

 indicatrix. 



39. The scalar invariants. We have now interpretations for 

 two fundamental invariants, G and h, and the expressions of these 

 invariants in terms of the coefficients «,/ and y,y. The indicatrix and 

 its position relative to the origin require for their determination, 

 apart from the rotation in space, five invariant scalars as remarked 

 by Levi (see note 27). The dyadic 4> has of course three invariants 

 ^s, ^2s, ^3 which are the coefficients in the characteristic equation 

 for $. Of these the last is $3 = a^dJ^xSxh)'- as in (101). The geo- 

 metrical meaning of <I>3 is, except for a factor, the square of the volume of 

 the cone intercepted by the plane of the indicatrix from the infinite surface 

 of Cone I. Except for a factor this is Levi's invariant A5; 4>s is his Ai; 

 and h his A2. 



