ROTATIONS IN HYPERSPACE. 685 



is equal in length to its projection on M2. On the surface generated 

 by the path curves of the group the circles for which vii = m-^ and 

 nil = — irvy are orthogonal. Hence the directions of the circular 

 sections form two orthogonal pairs. The center of mean curvature 

 bisects the curvature segment. 



Rotations in 5-space give nothing new since the path curves will lie 

 in the 4-space perpendicular to the fixed axis of the rotation and pass- 

 ing through the given point. 



9. Rotations in 6-space leaving the same three mutually 

 perpendicular planes, invariant. We will next consider the case 

 of 6-space in detail before generalizing. Evidently these transfor- 

 mations form a group. The directions which a point can move by 

 the various transformations of the group form a 3-space. These 

 directions are defined by 



(48) ^=T= ('"i^^^i + ^«2il/2 + m^M^) • ''■^ 



as CIS 



T is then a unit vector tangent to the curve given by a particular set of 

 values of vii, 1112, mg. Squaring (48) we get 



(49) (^J= im'iMvry + m,-^(M2-ry + m^Mz-rf 



which is constant for given values of ??h, '}m, ms since (Mi-r), (Mo-r), 

 (Ms-r) are of constant length. The curvature of the path curves is 



^ ' m,KM, ■ rY + m^(M2 • rY + m,'{Mz • r^ 



This shows that the end of the curvature vectors lie in a plane deter- 

 mined by the projections of /• on the three planes Mi, Mo, M3. We 

 see that for real values of vii, mo, ms that is for real directions through 

 the given point, the points lie inside this triangle, which we will call 

 the curvature triangle. To each point in the curvature triangle 

 corresponds four sets of values of vh, mo, viz, that is, four curves 

 through the point. Each of these curves have the same curvature. 



dr dr' 

 The angle between two directions — and -— is given b\^ the formula 



ds ds 



,,,. dr dr' mimi'i^IvrY -{- mimiKM^-rY + vizmz^iMz-rY 



(ol) — •— = 



ds ds Vm,'(Mi • rY + m^HMo • rf + vizKMz • rf 



-^m.^'-iMvrf + mo'^-iMo-rf + mz^\Mz-rY 



