116 PROCEEDINGS OF THE AMERICAN ACADEMY 



Three Rectangular Planes. 

 I=M', X-\-A : N 



J= M : r+ B : N' 

 K=M: Z-\- C : N 



One Solid. 

 V= M : J^ 



Unity. 

 l = 3f:M+ A:A-\-B:B-]- C: O 



+ iV^:iV^4-X:X+ T: Y -\- Z : Z 



This unity might be omitted. 



The relation of the two multij)lications is exceedingly interesting. 

 The system seems to me more suitable to three dimensional space, and 

 also more natural than tliat of quaternions. The simplification of 

 mechanical formulas is striking, but not more than quaternions would 

 effect, that I see. 



By means of eight rotations through two-thirds of a circumference, 

 around four symmetrically placed axes, together with unity, all distor- 

 tions of a particle would be represented linearly. I have therefore 

 thouffht of the nine-fold aljjebra thus resulting. 



