ROTATIONS IN SPACE OF EVEN DIMENSIONS. 159 



id represents a rotation through 90° and stretching of mochilus d, so 

 I-M represents a rotation through 90° in each of the fixed planes and 

 stretching in each of modulus equal to the angle of rotation. 



In the study of commutative rotations it is found useful to define 

 a new product of two-vectors which we call the star product. The 

 star product of 3/i and 3/2 is defined as the vector of the dyadic 



The cross product of two one-vectors in three dimensions is again a 

 one-vector. No such product of one-vectors in higher space can, 

 however, be defined. The star product combines two-vectors in three 

 dimensions in the same way that the cross product combines one- 

 vectors. It is however definable in space of any number of dimensions, 

 the product being always a two-vector. It may therefore be con- 

 sidered a generalization in higher space of the Gibbs cross product. 

 In four dimensions, if ilf 1 and M2 are unit simple two-vectors and 

 TTi and TTo perpendicular to and cutting both the star product is 



• 



3/1 * M-i = cos do TTi + cos di TT'i, 



di and Q2 being the angles intercepted in tti and tto between Mi and Mo. 

 If the rotations in the fixed planes are all through the same angle 6 

 there is a fixed plane through each vector of space and all vectors are 

 rotated through the same angle. These equiangular rotations have 

 characteristic eciuations of order two. They are analogous to the 

 rotations produced by quaternion multiplication ^^ and are of two 

 types corresponding to quaternions referred to right and left hand 

 axes. In four dimensions equiangular rotations of opposite types are 

 commutative and every rotation is the product of eciuiangular rota- 

 tions of opposite types. 



12 H. B. Phillips, Application of quaternions to four dimensions. Johns 

 Hopkins University Circular. January 1905, page 8. 



In four dimensions a quaternion may be considered as a vector referred to 

 system of four mutually perpendicular axes, one being the axis of reals, the 

 other three the axes on which i, j, k lie. The product of two such vectors a and 

 ,3 is a generalization of the product of two complex numbers, the result being 

 a vector which makes with each of the given vectors the same angle that the 

 other makes with the axis of reals. Two vectors are thus determined one 

 being a;3 the other [:1a. 



