82 Proceedings of the Royal Society of Edinburgh. [Sess. 
We find that it obeys the commutative law and that its square is negative 
unity. 
So ijkl = 0 5 m , 
- Q b ijkl = m . 
Hence, substituting for l , from the previous condition, we get 
Q 5 0 4 = m , 
and readily prove 
= - 0 4 12 5 . 
The general conditions are that O w is commutative with linear vectors where 
n is odd, and non-commutative with them where n is even ; while its square 
is negative unity when n has the form (4/* 4-1) or (4^ + 2), and is positive 
unity when n has the form (4^ + 3) or (4^ + 4), /u being an integer, zero 
included. 
In all cases in which our observations are upon directed phenomena 
occurring in tridimensional space, but which are actually, or merely de- 
scriptively, to be regarded as influenced by the existence of that space in 
space of a higher order, the appropriate algebra to be used in their 
investigation is that of quaternions with the addition of the symbol O of 
the higher space involved. 
(Added February 25, 1920.) 
It is not necessary to use as above the Hamilton-Clifford extension 
of quaternions in the treatment of relativity problems. Hamilton’s bi- 
quaternions, so exquisitely developed in his tremendous Lecture VII, are 
directly applicable (see Silberstein’s Theory of Relativity). 
(Issued separately May 29 , 1920 .) 
