352 WILSON AND MOORE. 



into a = and [^ = 0. The principal directions corresponding to fi = 

 become the true principal directions while those corresponding to 

 a = become the asymptotic directions. In the four dimensional 

 case when at the indicatrix reduces to a linear segment (not passing 

 through 0) the condition a«|X = breaks up into |J- = and a'|JL = 

 with \i- 9^ 0. The directions for which |Ji = correspond to the extremi- 

 ties of the segment, which may be called the true principal directions 

 and are perpendicular, while the directions for which a«|X = (jJi" ?^ 0) 

 correspond to the directions which may be called asymptotic from 

 analogy. These directions may be real coincident or imaginary, 

 but in any case are bisected by the principal directions, since the two 

 coincident points in which a line cuts the linear segment correspond 

 to values of 6 respectively greater than and less than those for which 

 JL = by equal amounts. 



Definition 2. There are other ways of generalizing the principal 

 directions to higher dimensions. For ordinary principal directions 

 JJL = 0, that is, |iL2 has a minimum. The lines for which [J-^ is a maximum 

 corresponds to the bisectors of the principal directions. If we desire 

 we can define as principal directions those for which ^ is a minimum 

 or maximum (it is not important to distinguish between the two 

 extremes when the indicatrix does not degenerate). Then the prin- 

 cipal directions on the surface would be four in number, spaced equally 

 at angles of 45° around the point on the surface and corresponding 

 to the axes of the indicatrix. 



Definition 3. There is another property which will define lines of 

 curvature in ordinary space on all but minimal surfaces. If any 

 direction X be drawn on the surface at a point, the change of the 

 normal dn along that line has a definite direction. It is possible to 

 find another direction X' such that the change dti' of the normal along 

 that direction is perpendicular to dn. In general X' is not perpendicu- 

 lar to X. But for the principal directions X' and X are perpendicular. 

 Thus: The principal directions are the pair of perpendicular directions 

 for which the differential changes of the normal are also perpendicular. 

 We shall examine the value of this (third) definition for principal 

 directions in any number of dimensions. We may consider the 

 equation 



^'".'f=0, (119) 



which will connect two directions on the surface. Instead of setting 

 up the relation in general we shall use formulas (73), (80), (82), (84) 



