222 Proceedings of Royal Society of Eclinlicrgli. [sess. 
we follow O’Brien, Heaviside, and Macfarlane, and make 
equal each to + 1, we get an algebra with non-associative products, 
an algebra which certainly is not quaternions.^ Heaviside is 
apparently unaware of the non-associative beauties of his system, 
which he believes “to represent what the physicist wants;” for he says, 
much to the credit of the Philosopldcal Transactions, that his system 
is “ simply the elements of Quaternions without the quaternions, with 
the notation simplified to the uttermost, and with the very incon- 
venient minus sign before scalar products done away with.” {Phil. 
Trans., vol. clxxxiii., 1892, p. 428.) 
(7.) An important operator in quaternions is V or + /SSg + 
where are space-differentiations along the mutually rectangular 
directions of the unit vectors a /3 y. This operator has been called 
Hablaf by Eobertson Smith, and Maxwell and Tait have adopted 
* Macfarlane fully recognises this, and discusses at considerable length his 
two products of one arrangement of three vectors, and his five products of one 
arrangement of four vectors. If we represent the vectors of the new system 
by abed . . . and the Hamiltonian vectors by a(^yl . . ., we see at once that 
ab= -K(ajS)= -i8«. Hence 
(ab)c= - yu(i and a(bc)= - 
and the reason why (ab)c and a(bc) have different values is evident. But, by 
a similar process, 
a( 2 y= {a(h)y — — c(ab) 
and u(iy = u[(hy)= -(bc)a 
so that, in this non-associative system, each association of one arrangement is 
equal to a particular association of one other arrangement. In quaternions 
the different values of the product of three vectors are got by permutation and 
by permutation only ; in this system they are got partly by different asso- 
ciations, and partly by permutation. But each system gives precisely the 
same number of different products. 
Similarly, although there are five different products — ((ab)c)d, (a(bc))d, 
(ab)(cd), a((bc)d), a(b(cd)) — got from the one arrangement, any one of these 
has its four equivalents which are particular associations of other arrange- 
ments. Thus it maybe shown that (ab)(cd)-=((ad)b)c = (d(ba))c = b((dc)a) 
= b(c(ad)). We commend this jungle, which the forsaker of the Hamiltonian 
track cannot escape if he only go far enough, to the careful consideration of 
Heaviside. It is small wonder that Grassmann (whose Ausdehnungslehre of 
1862 does not hint at the possibility of putting i^= -1) never found leisure 
to apply his own system to angles in space. He never formed the geometric 
conception of a quaternion, his vector quotients being quite other things. 
t Heaviside regards this name as “ludicrously inefficient,” whatever that 
may mean. Some name is sorely needed. To invert Delta gives an awkward 
