JOURNAL 



OF THE 



WASHINGTON ACADEMY OF SCIENCES 



Vol. 9 JANUARY 19, 19 19 No. 2 



MATHEMATICS. — Note on rotations in hyperspace. Edwin 

 BiDWELL Wilson, Massachusetts Institute of Technology. 



In a recent paper C. L. E. Moore ^ has discussed rotations in 

 hyperspace, by treating first the resolution of a complex 2 -vector 

 M into mutually completely perpendicular simple 2 -vectors or 

 planes. The problem is of sufficient interest, perhaps, to justify 

 my sketching very briefly the method by which I attacked the 

 problem of discussing rotations in hyperspace in a paper read 

 before the American Mathematical Society- over ten years 

 ago, but never published. My method was founded on the 

 general reduction of the homogeneous strain to canonical form, 

 and in particular on the work subsequently printed in this Jour- 

 nal.^ It is there shown that a real strain about a fixed point 

 has four types of transformation as possibiHties. 



1. The tonic. That is a stretching along a fixed direction. 

 This corresponds to a real root of the characteristic equation, 

 and to each real root there is at least one such direction in space. ^ 



2. The shear. This arises only when two or more real roots 

 of the characteristic equation are equal, and consists in the 



^ Moore, C. L. E. Rotations in hyperspace. Proc. Amer. Acad. 53: 651-694. 

 1918. 



^ See, Bull. Amer. Math. Soc. 13: 265. 1907. 



^ Wilson, Edwin B. Note on multiple algebra; the reduction of real dyadics and 

 the classification of real homogeneous strains. Journ. Wash. Acad. Sci. 7: 173-177. 



1917- 



* The multiplier may be negative, «. e., the stretching may be accompanied by a 

 reversal of direction; nevertheless for analytical reasons, we may speak of a fixed 

 "direction." 



