98 Mr Brill, On the Generalization of [Feb. 24, 



(2) On the Generalization of certain Properties of the Tetra- 

 hedron. By J. Brill, M.A., St John's College. 



1. In a paper 1 published in the Messenger of Mathematics for 

 last August, I obtained a generalization of some of the properties 

 of the plane triangle with the aid of ordinary algebra. I now 

 propose to utilize the theory of binary matrices to obtain a 

 generalization of the corresponding properties of the Tetrahedron. 

 The methods used will be found to bear a close analogy to those 

 used in a paper printed in the sixth volume of the Proceedings of 

 the Society 2 . 



2. In order to facilitate the working out of the subject, we 

 will adapt the notation used by Hamilton in his theory of quater- 

 nions to the theory of binary matrices, which is really the theory 

 of quaternions in a somewhat different form 3 . 



Let m be a binary matrix whose characteristic equation is 



m 2 — 2\m + fi= (1). 



We will define the operators U and W with the aid of the 

 equations 



Um = \, ( Wnif = fi, 

 where, for the sake of definiteness, we suppose that Wm represents 

 the positive value of V fi. 



The operator K will be defined by the equation 

 Km = 2\ — m, 

 so that we have 



m + Km = 2 Um (2). 



We also have 



m . Km = Km . m = 2\m — m" = fi = ( Wm) 2 (3). 



In addition to the above operators, we will introduce an 

 operator V defined by the equation 



m = Um + Vm, 

 so that we have 



Km = Um — Vm (4), 



1 "Note on the Application of Analysis to Geometry." Messenger, xxv. 49 — 59. 



2 " A New Geometrical Interpretation of the Quaternion Analysis." Proc. Canib. 

 Phil. Soc, vi. 156—169. 



3 It will be seen that K and V correspond to Hamilton's K and V, while U 

 corresponds to his S, and W to his T. Hamilton has no symbol corresponding 

 toL. 



