566 
Proceedings of the Royal Society of Edinburgh. [Sess. 
either i 2 = — 1 or that the sign of i 2 is ambiguous. Must we or must we 
not assert that economy of thought demands that t 2 = 1 ? Must we or must 
we not assert that the same economy demands the simplest possible three- 
dimensional geometrical method ? I am really quite doubtful, but rather 
incline to choose £ 2 = 1. This makes {l^l^) 2 — —1 and so prevents us from 
regarding q* 2 * 3 as a real scalar. Nevertheless, even with £ 2 = 1 and n = 3, 
Quaternions is still a real particular symmetrical case of Multenions; 
quaternions is the calculus of multenions q, r, ... . such that S-^q = S s q = 0, 
S 1 r = S 3 r = 0, .... And further, we may still, if we please, assert that 
£ 1 £ 2 £ 3 is a scalar, namely the scalar J( — 1). Thus with n = 3, i 2 = 1, 
~ scalar, a multenion becomes identical in properties with Hamilton’s 
bi-quaternion q-\-r J( — 1) where q and r are real quaternions. 
At the same time I do not recommend this course when l 2 — 1, n = S: 
Instead, I recommend that iy 2 i 3 be not regarded as a scalar but as an 
independent real multit. Further, I recommend the following as the 
standard geometrical interpretation. Identify Actors with vector lines, 
and therefore second order multenions with vector areas, and third order 
multenions with volumes. [At the same time remember that Actors 
may be identiAed with vector areas, and second order multenions with 
vector lines.] 
I think there can be no question that the geometrical method we thus 
get is distinctly inferior to the quaternion method ; but for all that, if we 
must choose between i 2 == +1 and i 2 — —1, 1 am inclined to choose i 2 = + 1 in 
spite of its inferior geometry. 
Perhaps the best plan is that adopted in this paper of leaving the sign 
of l 2 doubtful. If we make our primary * 2 = .l, the complementary replace- 
ment produces cases where t 2 = — I. 
The treatment in § 4 above of q~ x is not complete. I have failed to 
make it reasonably simple. The reader will probably And little diAiculty 
in supplying proofs of the assertions now to be made. 
What is to be desired is some simple function Xg of q (simple in the 
sense that it can be easily written down when q is given) such that gXg is 
a scalar which is only zero when q~ x is inAnite. 
If q is a given multenion, r an arbitrary one, and x any scalar, the 
multilinity (p given by 
cf> r = ±(qr + rq) + %x(qr-rq) . . . (1) 
has the property that for all positive integral values of a 
* a l = 2 ° ...... ( 2 ) 
Hence the 2 w -tic satisAed by <p is also satisAed by q. Also in general 
