SURFACES IN HYPERSPACE. 353 



to express the condition that, for two perpendicular directions, the 

 differential planes are orthogonal. We find 



^ . 'i^ = (ax-n + |xjjL).(|jLxTi + |xp) = a. HI + p.jj. = 2h'[i. = 0. 

 ds ds 



(120) 



The condition for principal directions is therefore noio h*|Ji = 0; 

 the directions on the surface are those for which jA is perpendicular to h. 

 There is one line in the plane of the indicatrix that satisfies this condi- 

 tion on }JL, namely the intersection of the plane of the indicatrix with 

 the plane through the end of h perpendicular to h. Two perpendicu 

 lar directions on the surface are determined by the two opposite values 

 of fJL. Hence: By definition 3 there are just two principal directions 

 through each point of the surface, and these are orthogonal. On a surface 

 for which |A = these two directions coincide with those previously 

 called principal. In case h = 0, the condition is satisfied for any 

 direction on the surface, and in case h is not zero but is along the axis 

 of Cone I, the condition is also satisfied identically. This last case 

 may perhaps be Kkened to an umbilic in ordinary surface theory — 

 for at an umbilic the principal directions are indeterminate. 



The expression clM/ds = axT| + |x|i gives 



(dM/dsY = a2 + \i:^ = h- + 52 + jx^ + 2h-8. 



As the expressions h- and [i-- + ^- are invariants, the maximum and 

 minimum values of (dM/ds)'-^ will fall where 5 has the greatest (posi- 

 tive or negative) projection on h, that is at the point of tangency of 

 planes tangent to the indicatrix and perpendicular to h, and for this 

 condition h*|JL = 0. The principal directions (definition 3) are therefore 

 those for which d'M./ds is a maximum or minimum hi magnitude, as in 

 the ordinary three dimensional case. It may reasonably be asked 

 whether such a condition as the maximum or minimum of d'M./ds 

 in magnitude is not more intimately connected with the surface than 

 the similar conditions on the curvature of a normal section. Unfor- 

 tunately the condition breaks down for the case h = 0, but there are 

 important theorems on principal directions in the three dimensional 

 case which suggest that h = is a really exceptional case.** 



It is not difficult to make a choice between the three generalizations 



44 See, for example, Eisenhart, Differential Geometry, p. 143. 



