550 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Law A is not assumed to hold with the general fictorlinity replacement. 
If Law A is imposed , then the fictorlinity replacement ( when real) reduces 
to a combination of a unireplacement as defined above and the rigid 
replacement e w ( )e~ U} . I shall not formally prove this below, but it will be 
quite obvious from the general argument. The fact shows that the above 
definition of unireplacement covers the most general case for which R£ = 1 ; 
so that this last may be taken as the definition of a unireplacement, though 
I have thought it clearer to use the full description of properties of R u as 
its definition. 
The fictorlinity replacement may be regarded as the generalisation of 
homogeneous strain to Euclidean space of n dimensions. In three dimen- 
sions if a body is homogeneously strained : — 
(1) Any vector line \ of it becomes (is replaced by) (p\ where cp is a 
general vectorlinity. 
(2) Any vector area cr = VA i a of it becomes (is replaced by) [0]0 /- V, 
though this has to be modified or further interpreted when [<^>] = 0. 
(3) Any volume, v, of it becomes (is replaced by) [c p]v . 
The fictorlinity replacement similarly shows for n dimensions how 
every region of c dimensions, included, is strained, and furnishes simple 
expressions for the strained region. [“ Region ” is the generalisation of the 
three terms, (1) vector line, (2) vector area, (3) volume.] 
Let <h, 'SE r be (what may be called ordinal) multilinities having the 
property that, acting on a multenion of order c, they convert it into 
another multenion of order c ; c being any positive integer. Thus 
&$& = $ = (say)* / g x 
^S c g = S c ^g = ^ c g (say) J ' ‘ ' . ' 
Thus an ordinal multinity is commutative with S c . We clearly have 
$ = 3> 0 + ^ ^ ^ + (9) 
w = (10) 
Let (p , be any two given fictorlinities and let {^>}, {fi} be multilinities 
defined by 
{$}£= l(<Py)m\ V M .... ( 11 ) 
the second form being given by (13) § 7. We shall eventually show that 
{ <p) may always be regarded as a replacement (with qualifications as to 
meaning similar to those explained for the rigid replacement), and that this 
replacement is the most general form of fictorlinity proplacement as already 
described. Meanwhile, we regard it merely as a mutilinity whose pro- 
perties are to be investigated. 
