516 
Proceedings of the Royal Society of Edinburgh. [Sess. 
formulae. I have not been able to obtain a similarly satisfactory 
generalisation of the formulae 
Ya/3y = aS /3y - /3Sya + ySa /3 
Y a Y [3y = - (3Sya + ySa/3, 
but the multenion formulae immediately suggested are true, namely, 
S x a/3y = aS/3y — /3Sya + ySa /3 ) 
Si(aS 2 /3y) = — /3Sya + ySa/3 J ^ 
[These are proved below at end of § 5 and the desired “ satisfactory 
generalisation ” is found, that is, it is enunciated, but not proved.] 
(6) may be put in a variety of other suggestive forms such as (multiply 
by a’ 1 ) 
2S & _iag & = a q b — ( - ) b q b a, 2S 6+1 a q b = aq b + ( - fq^a. / g^ 
(S 6 _i + S b+1 )aq b — aq b , (S 6+1 - S b-i)aq b = ( — ) b q b a f 
We now pass to the main purpose of this section. 
Replacements in general will be considered in a subsequent section. 
Meanwhile it may be said that the term is meant to connote two different 
trains of mathematical ideas : — (1) the kinematics! ideas of displacement ,, 
whether (a) that of a rigid body, or (6) that of a deformable system called 
homogeneous strain, or (c) this combined with perversion such as is. 
produced by a plane mirror, or (d) strain in general, including such strain 
perversions ; (2) the algebraic ideas involved in Grassmann’s various species 
of “variation” and my own extensions of them in Octonions. The last 
are all of the nature of replacing certain given symbols by others according 
to imposed rules. 
The meaning of the (in some respects) simplest of the replacements, 
the rigid replacement q( )q~ 1 , to be considered, is probably sufficiently 
explained in the following enunciation. 
(1) The original Jictits i v i 2 , .... of the Jictorplex *p 2 . ... i n may for all 
the purposes whatsoever of Laws (1) to (4), § 2, and Law A, § 3, be replaced 
by n other multenions \ v \ 2 , .... \ n where 
~ q L \q •> h 2 = qi 2 q ^n~q L nq .... (9)* 
where q is any given multenion for which q -1 is finite. 
(2) By such a replacement any multenion r of the jictorplex is 
replaced by qrq - . 
Law (1), § 2, merely states that N fictits q , i 2 , . . . . are given, and that 
the laws of ordinary algebra, with an exception, are to apply. 
Law (2). A scalar is commutative with the multenion 2(^ 0 + ^ 1 X 1 + . . . A 
(a/ 0 + a/ 1 A 1 + as with every other multenion. 
