ROTATIONS IN HYPERSPACE. 691 



infinite number of point pairs corresponding to perpendicular direc- 

 tions. For, any point on tlie line can be taken as the center of a 

 curvature ellipse which touches the three sides of ABC. The two 

 points in which these ellipses cut the line correspond to perpendicular 

 directions. The points corresponding to the directions perpendicular 

 to these pairs will all lie on a line parallel to the given line. 



Proceeding as in 4-space the differential equations of the path 

 curves are found to be 



, d.ri dxo (1x5 d.n 



at at (it at 



The path curves will then all lie in the variety V^^ of order 8 



(59) .xr + .ro- = o^ x^' + .r4- = Jf; x,' + x,' = c^, 



where a, b, c are determined so that the curves all pass through the 

 given point. If lUi = ^nij the resulting path curves will lie in a 

 4-space but if nu = kuij (k ?^ ±1) this is not the case. If mi = 

 =>=mo = ±???3 the resulting path curves are plane curves. The 

 argument is the same as that given in 4-space. 

 The parametric equations of Vs^ are 



Xi = a cos u, X2 = a sin ii, 

 xs = b cos V, Xi = b sin v, 

 X5 = c cos IV, .Te = c sin w. 



The element of arc is 



(60) ' ds^ = a-div" -f b'-dv" + cHii" 



The variety can therefore be developed on a plane 3-space. The 

 path curves have curvature lying in the normal 3-space (the 3-space 

 determined by the surface point and the curvature triangle) and are 

 therefore geodesies. That they are geodesies can be shown directly 

 from the above equations as was done for rotations in 4-space. A 

 linear relation among the m's will give a surface which is left invariant 

 by a one parameter family of rotations. This is then a geodesic 

 surface of the variety IV. Furthermore the normal 4-space to this 

 surface must contain the normal 3-space to V^ and the ends of the 

 curvature vectors of the pencil of geodesies passing through a given 

 point will trace out a conic lying in the curvature triangle and since 

 this lies in the normal to the surface these path curves must be geodes- 

 ies on the surface K. A linear relation in the m's means a linear 



