158 PHILLIPS AND MOORE. 



tion of a complex two-vector into the sum of completely perpendicu- 

 lar simple two-vectors or planes. Gibbs' vector analysis and dyadics 

 form a foundation of this work. E. B. Wilson ^ (1919) published a 

 note outlining a discussion of rotations founded on the properties of 

 the characteristic equation. 



The group side of the subject has received more attention than the 

 other two. Bemporad ^ (1904) returning to the original method of 

 Jordan applied the general theory of Lie to determine the different 

 types of groups of motions in three and four dimensions. E. E. Levi ^° 

 (1905) considered the | 7i{'n -\- 1) parametered group of motions in 

 ?i-space. He treated the two and three parametered subgroups of 

 the general group. The paper is intimately connected with a paper 

 by Bianchi on generalized motion. C. L. E. Moore ^^ (1918) discussed 

 motions in 7i-space from the point of ^'iew of the infinitesimal trans- 

 fomiation. 



In this paper we use the Gibbs dyadic to discuss rotations in a 

 space of 2n dimensions, particularly in four dimensions. It is shown 

 first that any such rotation is the product of rotations in n fixed 

 planes. The dyadic representing the general rotation is the sum of 

 d^^adics representing the rotations in these fixed planes. If we asso- 

 ciate with each plane the two-vector in that plane equal in magnitude 

 to the angle of rotation, the sum of such two-vectors for all the fixed 

 planes is a complex two-vector M which plays a role analogous to 

 that of the one-vector, equal to the angle of rotation, along the axis 

 of a three dimensional rotation. A rotation in 2n dimensions may 

 thus be described as a rotation through a two-vector angle M. Just 

 ;as a rotation in the plane can be expressed in the fonii 



A b I— I 



where S and H' are corresponding complex variables, so the rotation 

 in 2n dimensions can be expressed in the form 



r' = c^-^r 



where r' and r are corresponding one-vectors. Furthermore, just as 



8 Note on Rotations in Hyperspace. Journal, Washington Acad. Sci. 9, 

 1919. 



9 Sui gruppe dei movimenti. Annali della r. scuola normale sup. di Pisa, 8, 

 1904. 



10 Sui gruppi di movimenti. Atti dei Lincei, series 5, 14, part 1, 1905. 



11 Motions in Hyperspace. Annals of Math, second series, 19, 1918. 



