QUADRATIC VECTORS. 405 



tangent pl.anc. If dp is parallel to p, dv vanishes, because we are 

 dealing with cones; it is sufficient, therefore, to consider any other 

 one direction of dp in the tangent plane, preferably the direction Vpv, 

 the direction of maximum curvature. If we write dv = xdp, Vpv is 

 an axis of the linear vector function x. and xVp^ — gVpv, where g 

 is the maximum curvature of a normal section of the cone.^* But this 

 curvature is equal to the "divergence " of the unit vector v at the 

 point on the cone. For, by definition of divergence, 



div V = — SVv = — Svxv — Siixu — Sexe, (100) 



where u and e are unit vectors along p and Vpv, respectively. But 

 Svxv vanishes, because the differential of a unit vector is always 

 perpendicular to the unit vector itself. And x^ vanishes because we 

 deal with cones. As above, xe = g^, giving, (because e^ = — 1), 



g = - SVv . (101) 



The necessary and sufficient conditions for a triple axis may ac- 

 cordingly be stated: if /5 is a triple axis of Fp, a unit vector normal to 

 the cone (99) and its divergence have at most one determinate 

 direction and one determinate numerical value, respectively, inde- 

 pendent of X. 



18. The direction of the normal is found by operating with V on 

 S\pFp. If we put, for convenience 



so that 



we have, from (101), 



a = VSKpFp (102) 



(103) 



TV 



g= -SV 



_ a^SVo' -\- S(X(p(T 



(104) 



14 For a more detailed examination of x, see Phil. Mag., June, 1902, p. 576, 

 and Feb., 190-3, p. 187. s , , , P , 



