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XXXIV.—Algebra after Hamilton, or Multenions.
Published online by Cambridge University Press: 15 September 2014
Summary
Ever since I have learnt something of the meanings of Grassmann's Ausdehnungslehre, and have at the same time learnt to regard the beauties of that system with something akin to awe, I have been persuaded that on the lines of Quaternion Algebra there is to be built a system very much like the Ausdehnungslehre, but an improvement thereon. Of course it will be matter for differing opinions whether what I call Multenions is really an improvement on the Ausdehnungslehre. I here record my own personal opinion that it is.
I do not suppose that anybody will maintain that a multitude of different kinds of multiplication within the bounds of one method can be regarded as anything but a blemish,—a blemish that may be justified by necessity and utility. The Ausdehnungslehre seems to me to have this blemish, and Multenions not to have it. Whether along with the absence of the blemish there becomes present an additional difficulty of manipulation is questionable. I have not found it so, but this may be due to the fact that I have so long been in the habit of thinking through the quaternion machinery.
I think these general remarks are all that can be usefully set down as a summary. The paper itself appears to be too condensed to admit of a true summary other than a mere table of contents.
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- Copyright © Royal Society of Edinburgh 1908
References
page 517 note * Invariably when n is even, in general when n is odd, by the next footnote below.
page 520 note * I am now (1908) able to show that when n is even q−1 is finite. If q−1=∞ let qq0=0=q0q where q0 is not zero. Since qp=p′q we have that qpq0=0 for all multenion values of p. In the Supplement below, between equations (44) and (45), it is proved that when n is even this is never true when both q and q0 differ from zero. Hence g−1 is finite. The case of n odd still remains incomplete, qpq0 may be zero for all multenion values of p when n is odd, as we see by putting 
page 524 note * I find (April 1908) that (12) below may be much more simply proved thus: The “transference” of 
may be effected in x steps by transferring each fictor 
of 
in succession; and we have to prove the theorem for one such transference (say of ξ1) only. The last is quite simply proved by expressing all the fictors in fictits c, which constitute the fictorplex 
[When the fictors 
are not independent the theorem obviously tabes the form 0=0.]
page 529 note * There is another and perhaps simpler method of dealing with what follows, by means of linities. Let φ be given by α1,α2,…. or the latter by the former by the equation
so that α −1=ϕt 1 etc. ϕ−1 is not infinite. Then ᾱ−1=ϕ1−1‘I’ ᾱ−2=ϕ1−1t2 etc., and in the notation of § 8 below αc(c)=ϕct(c) etc., ᾱc(c)=ϕc1−1t(c) etc.—[Note added April 1908.]
page 537 note * There is here a most unfortunate oversight in the notation. In § 8 below, a series of multilinities φ 0, φ 1φ 2….φ n of a quite fundamental kind are defined in terms of φ; φ 1=φ and φ c is of c dimensions in φ. The oversight is that in this series φn is the scalar here denoted by [φ]. φn is a far more expressive symbol than [φ], and should, throughout the paper, be read instead of [φ].—[Note added April 1908.]
page 559 note * Mischievous because it hides some of the inherent simplicities of the method.
page 560 note * I find (April 1908) on p. 116 of Sci. Papers that Professor Gibbs recommended identically this notation in 1886.
page 584 note * I have prepared this brief epitome of what seem to be the salient points, to facilitate reference.—C. G. Knott.