ROTATIONS IN HYPERSPACE. 

 By C. L. E. Moore. 



Received, January 2, 1918. Presented, April 8, 1918. 



In this paper three problems are discussed. First, the resohition 

 of a complex 2-vector M, in space oi 2 p dimensions into the sum of p 

 mutually completely perpendicular simple 2-vectors or planes. It is 

 shown that this can always be done and is in general unique. But 

 if M satisfies certain product relations the resolution can be effected 

 in an infinite number of ways. In four dimensions this relation is 

 equivalent to saying that M is what Whitehead ^ calls self-supple- 

 mentary. In this case the resolution can be effected in oo- different 

 ways. 



Second, the application of the preceding to show that a rotation in 

 a space of 2 ;; dimensions leaves p mutually completely perpendicular 

 planes invariant. In general these are the only invariant planes. 

 But in case the rates of rotation in the p invariant planes are the same 

 or differ only in sign there are an infinite number of invariant planes 

 which can be arranged in sets of p which are mutually completely 

 perpendicular. In 4-space in case the rates of rotation in two com- 

 pletely perpendicular invariant planes have the same magnitude 

 there are oo- invariant planes which are completely perpendicular in 

 pairs. 



Third, the consideration of the variety Vp left invariant by all the 

 transformations leaving the same set of p planes invariant. It is 

 found that this variety is of order 2^". The path curves are curves of 

 constant curvature and first torsion and are geodesies on Vp. The 

 centers of curvature of all the path curves that pass through a given 

 point lie on a sphere of ;j-dimensions having the given point P and 

 the point of intersection of the p invariant planes as ends of a diame- 

 ter. Through each point pass 2^ path curves which are circles having 

 for center and OP for radius. The variety Vp can be developed on 

 a plane space of p dimensions. 



1 A treatise on Universal Algebra, page 292. 



