WILSON: NOTE ON ROTATIONS IN HYPERSPACE 27 



in which stretching took place. Next, if a real root is double, 

 there can be no shear corresponding to that double root, because 

 in the shear all lines through the origin in a fixed plane through 

 the fixed direction must change their direction, with the sole 

 exception of the fixed direction itself, and an angle cannot remain 

 always unchanged in the case of a shear. In like manner, it 

 may be seen that when an imaginary root is multiple, the shear- 

 ing terms must be absent if the strain reduces to a rotation. 



If there are a certain number of real roots equal to + 1 , a space 

 of the same number of dimensions is left absolutely unchanged. 

 If there are a certain number of roots — i , these may be paired 

 to represent a rotation of i8o° in a certain number of planes, 

 which will take care of all if the number of roots — i is even. 

 If the number is odd, there will be one direction left over, along 

 which directions are reversed. This case corresponds not to a 

 rotation, but to a reflection, and, consequently, the number of 

 roots — I must be even. The transformation of rotation, there- 

 fore, reduces to the identical transformation in a space of a cer- 

 tain number of dimensions, and to rotations through angles 

 of 1 80 ° or otherwise in a number of independent planes sufficient 

 to make up the total multiplicity of the hyperspace. 



The fact that there are no shearing terms in the dyadic or in 

 the matrix, $, means algebraically, that the equation of lowest 

 degree satisfied by the dyadic has only as many factors as there 

 are different roots in the characteristic equation, each factor 

 raised to the first power, namely: 



($ - !)($ -\- I)($-e'^I)($ — g-'"!) =0 



or 



($2 — I)($2 _ 2C0Se$ +1) =0 



It is necessary to point out that the different independent 

 spaces which correspond to the dift'erent roots of the equation 

 are perpendicular. Consider, for instance, any direction in a 

 space corresponding to -fi, and any direction in that corre- 

 sponding to — I. The transformation leaves the first direction 

 unchanged, and reverses the second, so that each angle is changed 

 into its supplement; and if angles are to remain unchanged, the 

 directions must be perpendicular. In like manner, there may 



