1907-8.] Algebra after Hamilton, or Multenions, §11. 
569 
SUPPLEMENT. 
The central proposition of this supplement, that a multenion may be 
regarded as a linity, was sent to the Royal Society, Edinburgh, in November 
1906. The present form was given in the correction of proofs, April 1908. 
As originally despatched, the final sentences of § 11 consisted of 
speculations as to the bearings of equations (7), (8), (9) of that section. 
If the paper were re-cast, the substance of the latter half of this supplement 
ought to occupy a prominent early position, say in § 2. 
I have now succeeded in simplifying the problem — given q, what is q~ l ? 
The hint came from Cayley’s § (c), p. 148 of Tait’s Quaternions, 3rd ed., 
where he shows that a quaternion is a linity of the second order. I am 
now able to show in the same sense that a multenion belonging to a 
multiplex of order 2m— 1, or of order 2m, is a linity of order 2 m , so that, as 
a particular consequence, it satisfies an identity of degree 2 m , instead of the 
22 m— i or 22 m con templated in § 11. 
I may explain that the reason for leaving unproved any theorem below 
whose treatment I have not succeeded in reducing to a simple form, is that 
I have hesitated to further amplify this already too long supplement. 
The linity theorems of § 7 above down to eq. (55), and also those of § 8, 
are true of linities in general, whether real or imaginary ; but the argument 
of § 7 from eq. (56) to the end depends on assuming the linities to be real. 
Corresponding theorems, when the restriction is removed, are by no means 
so simple. It is necessary, for application to a multenion as a linity, to 
consider now these generalisations, and some allied linity properties. 
We begin by developing the fundamental Grassmann theorem of 
equations (21), (22), (23), § 7 above, and assume the notation there explained. 
We have 
(<t>-a)\<f>-by .... =0 (15) 
where a, b, ... . are all different. Let the minimum degree identity 
corresponding to this be 
(^-ay^-by^-cy. ... =o . . . . (ie) 
so that g, h, etc. are positive integers, of which g the first is equal to or less 
than A, the second h equal to or less than B, and so on. Let g + , h4~, etc. 
stand for any positive integers equal to or greater than g, h, etc. respectively ; 
so that in particular we may put either g-\- =A etc., or g-\- =g etc. 
x being any symbol, expand the fraction 
1 /(x - a) g+ (x - b) h+ .... 
