SURFACES IN HYPERSPACE. 353 



to express the condition that, for two perpendicuhtr cHrections, the 

 differential planes are orthogonal. We find 



(IM ^ dm ^ ^^^^ ^ |x|JL) . (|JLx-n + |xp) = a.[Ji + p.jjL = 2h.|JL = 0. 



as as 



(120) 



The co)i(litioti for principal dircrtious is therefore vow h'\i- = 0; 

 the directions on the surface are those for which (i is perpendicuhir to h. 

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

 tion on |J-, 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 Aalues 

 of \i.. Hence: Bi/ definition 3 there arc just two principal directions 

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

 for which [^ = 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 likened to an vunbilic in ordinary surface theory — 

 for at an umbilic the principal directions are indeterminate. 

 The expression dM/ds = axT| + |xfJL gives 



(dm V/.?)2 = a2 + 1x2 = h'2 ^ 82 _|_ p,2 ^ 2h-5. 



As the expressions h' and p.- + 5- are invariants, the maximum and 

 minimum \alues of {dm/ds)~ will fall where 8 has the greatest (posi- 

 tiA'c 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'p. = 0. The principal directiotts {definition 3) are therefore 

 those for which dm/ds is a maximum or minimum in magnitude, as in 

 the ordinari/ three dimensional case. It may reasonably be asked 

 whether such a condition as the maximum or minimum of dm/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. 



