ROTATIONS IN HYPERSPACE. 681 



two vectors are perpendicular to each other. The vector Mi-r is 

 perpendicular to Mi-iMo-r) since they lie in completely perpendicular 

 planes. Therefore Mi • r is perpendicular to the plane of the curvature 

 vector. Similarly M^-r is also perpendicular to this plane. Hence, 

 The path curves of a giren point hi/ the transformations of the group of 

 rotations ivhich leave the same two completehj perpendicular planes 

 fixed are all tangent to a plane A, and have constant curvature and torsion. 

 The ends of the curvature vectors lie on a line cuttijig the two fixed planes. 

 The plane in which these curvature vectors lie is perpendicular to the 

 plane A. There arc four of the path curves ivhich are circles. The 

 tangents to two of these are perpendicular to each other and the tangents 

 to the four form a harmonic pencil. The centers of curvature lie on a 

 circle whose diameter is the line joining to the given point. The centers 

 of curvature of the real curves lie on one quadrant of this circle. 



As the vector r is rotated by (39) the plane A will envelope a surface 

 which will be left invariant by every transformation of the group. 

 To obtain the equations of this surface we will integrate the vector 

 differential equation (39). Let 



r = xih + .ToA-o + .TsA's + 3:4^:4) 

 Ml = k]2) Mo = A'34 



Then (39) becomes 



kidki + A-2cZA-2 + ksdks + kidxi = (jnikn + mok3i)-(xiki -\- Xoki 



+ 3^3^:3 + Xiki)dt 



= {viiXoki — viiXiko + moXik's — m^xzk4)dt 

 which is equivalent to the set of differential equations 



dxi dxz 



—- = miX2, -— = —mixi, 

 dt dt 



dxz dxi 



— - = miXi, — - = — ?722.r3. 



dt dt 



Dividing and integrating we obtain for the first integrals 



(43) .Ti2 + xi = a\ x,^ + .r4-- = b'. 



The constants a and b are so determined that the curve will pass 

 through the initial point. These then are the equations of the surface 

 left invariant by each transformation of the group. The planes A are 

 the tangent planes to the surface and the normal planes are those in 

 which the curvature vectors lie. These normal planes all pass through 



