ROTATIONS IN SPACE OF EVEN DIMENSIONS. 

 By H. B. Phillips and C. L. E. Moore. 



Received December 30, 1919. Presented February 11, 1920. 



Introduction. ^Motions in three-space have been studied from 

 three points of view, (a) the geometric in which the reasoning is pri- 

 marily synthetic; (b) the algebraic expressed in terms of quaternions 

 or Gibbs' dyadics ; (c) that of groups of motions which was started by 

 Jordan ^ (1884) and was continued by Lie and others. The literature 

 of motions in three-space is quite extensive. The study of motions 

 in ?i-space has not been nearly so extensive but has followed the 

 general lines of development of motions in three-space. One of the 

 first papers on the subject was by Cole ^ (1889). He discussed 

 rotations in four-space by means of a sort of generalized Plucker 

 coordinates. The principal result of the paper is that a rotation in 

 four-space leaves two and only two completely perpendicular planes 

 invariant. P. H. Schoute ^ (1891) by what might be called geometri- 

 cal argument discussed the general motion in n-space. The algebraic 

 method has not been used much so far as the authors of the present 

 paper know. Clifford * and Lipschitz ^ generalized ciuaternions and 

 thus established a transformation leaving the sum of squares invariant. 

 The papers on this subject have dealt mostly with the algebra and very 

 little ^nth the geometry of motions. 



The Gibbs' dyadic ^ is readily extended to n-space but its use for 

 studying motions has not been very extensive. C. L. E. Moore ^ 

 (1918) discussed rotations in hyperspace by treating first the resolu- 



1 Annali di mathematica, 1884. 



2 On rotations in space of four dimensions. American Journal, 12, 1889, 

 page 191. 



3 Le deplacement le plus general dans I'espace a n dimensions. Annales de 

 I'Ecole Polytechnique de Delft, 7, 1891. 



4 Applications of Grassmann's Extensive Algebra, American Jom-nal 1, 

 350, 1878. Papers, London, 1882. 



5 Untersuchungen ueber Smnmen von Quadraten. Bonn, Max Cohen & 

 Sohn, 1886. 



6 E. B. Wilson. On the theory of double products and strains in hyperspace. 

 Trans. Conn. Acad. Arts Sci. 14, 1-57, 1908. 



7 Rotations in Hyperspace. These Proceedings. 53, 1918. 



