80 
Proceedings of the Royal Society of Edinburgh. [Sess* 
IX. — On the Quaternionic System as the Algebra of the 
Relations of Physics and Relativity. By Professor W. 
Peddie. 
(MS. received February 2, 1920. Read February 2, 1920.) 
In a recently published paper ( Proc . R.S., Dec. 1919) Mr W. J. Johnston 
makes a brilliant use of Clifford’s extension of Hamilton’s quaternions to 
the case of four (or more) variables. He proceeds, by adding a fourth inde- 
pendent and real unit vector and making the corresponding tensor 
imaginary, to modify the quaternionic V so as to make it applicable to 
physical problems on the basis of relativity. In this respect his vectors 
are simply the direct extension of Hamilton’s bivectors. 
In an intensely illuminative paper on Generalised Relativity, immediately 
following the one referred to, Sir Joseph Larmor simplifies the expressions 
by making the added unit vector itself imaginary, so that its square 
becomes positive unity. 
Hamilton did not himself generalise his algebra, though no one could 
have done it more ably than he could have done. The latter part of his 
life was devoted to the development of quaternions in their relation to 
ordinary geometry of three dimensions. In these dimensions the vector 
product ijk has no meaning. The vector product ij can only indicate the 
plane ij, which is specifiable by k. Nothing is therefore gained by not 
identifying them and so putting ijk= — 1 . If we put ijk — — Cl, we find 
0 2 =+l, iVl = Vli, so that O has no properties different from those of 
positive unity. 
If we deal with a system of four unit vectors (i, j, k, l) and put 
ijkl= — O, we find that while, as in the case of three, Q 2 = +1, O is itself 
non-commutative with the other units. In fact Vli = — id. Thus we 
cannot identify O with positive unity. It possesses half of the properties 
of a unit vector, and can take the place of l in the vector p = xi + yj + zk -f ul. 
Thus ijkl=—Vl, ijk = Ql, l = Vlijk = — ijkVl, and p can be expressed as 
xi + yj + zk + uVlijk. It is true that relations such as ij = Qlk subsist, so 
that the 15 possible vector combinations are not all independent. Eight 
only remain. But these are sufficient for the statement of the result of 
any fourth dimensional effect upon a tridimensional system. The Hamilton- 
Clifford symbol O, added to the quaternionically conditioned system i,j, lc, is 
sufficient. 
