26 wii,son: note on rotations in hyperspace 



simplest case of the superposition upon the stretching of a shding 

 parallel to the fixed direction, and proportional to the distance 

 from the fixed direction, as is necessary if lines originally passing 

 through the origin are to remain lines passing through the origin. 

 The shear may be of various degrees of complexity; for example, 

 if there are three equal real roots there is a plane through the 

 fixed direction which is fixed, and in it the shear is simple ; whereas 

 the displacement of points not in this fixed plane is partly in 

 the fixed direction and partly in another direction parallel to 

 the fixed plane. (It is not necessary that there be shearing when 

 there are multiple roots. It is clear that two different direc- 

 tions in space may be fixed with the equal ratios of stretching 

 for those directions, and with the same ratio for all directions in 

 their plane.) 



3. Cyclotonic. This is a combination of stretching with ellip- 

 tic rotation in a plane. It arises when there is a pair of conjugate 

 imaginary roots in the characteristic equation. All lines are 

 stretched in a definite ratio, and are turned through a definite 

 "angle," provided angle be measured by the sectorial area in an 

 ellipse concentric with the fixed point instead of by the sectorial 

 area in a circle. 



4. The cyclotonic-shear. In the simplest case, where the 

 characteristic equation has a pair of conjugate imaginary roots, 

 each occurring twice, the transformation consists in a cyclotonic 

 change in each of two nonintersecting planes through the origin 

 combined with a displacement parallel to one of the pairs. For 

 conjugate imaginary roots of greater multiplicity various com- 

 plexities of cyclotonic-shear are possible. (It is not necessary, 

 however, that shear accompany multiple roots in this case any 

 more than in case the roots are real.) 



A rotation is a transformation in which length and an angle 

 remain unchanged. As length remains unchanged it follows that 

 all the real roots of the characteristic equation must be either 

 + 1 or — I ; for otherwise there would be at least one direction 

 in which stretching took place. Moreover, the magnitudes of 

 the imaginary roots must also be equal to +1, i- e., the roots 

 must be complex quantities or there would be at least one plane 



