608 Scientific Proceedings, Royal Dublin Society. 
Groups which resemble one another as to parts of them which are 
identical. Comparison with some cycle and non-cyclic combinations. 
There is a way in which groups which are not isomerides may 
be closely related, where the resemblance does not spring solely 
from an enantiomorphous relation, while, at the same time, such a 
relation is not excluded. Thus, instead of obtaining a number of 
different groups by changing the grouping of a given set of balls, 
we may fix upon some particular arrangement for a group, and 
then derive a series of related groups from it by substituting for 
one or more of its balls other different balls or complexes of balls, 
leaving the relative situation in the group of the remaining balls 
unaltered,! and waiving the question of the arrangement of the 
groups in the different assemblages. Groups obtained in this way 
will resemble one another as to all properties imparted solely by the 
identical portions common to them, but, in addition to this we have 
the interesting fact that under some conditions the number of dif- 
ferent groups obtainable from a given group by aspecified number 
of substitutions can be ascertained. For, if the general conditions 
are constant, or which amounts to the same thing, if we disregard 
any change of arrangement brought about by changes in these 
conditions, it is evident : 
1. That the substitution for a particular ball in each group of 
a certain different ball or rigid complex will modify the conditions 
of equilibrium of an assemblage of similar groups im a definite 
manner determined by the action of the fundamental law of closest- 
packing, so that the position of the substituted ball or complex 
with respect to the remainder of the group may be regarded as 
fixed, and this will still apply if the substituted ball becomes 
attached to its group. 
2. That when two or more of the same sort of original balls — 
are exchanged for others, and some of the balls left resemble those 
removed both in their nature and their situation, the exchange may 
be effected in a definite number of different ways depending on the 
nature of the grouping, the number of balls removed, and the 
number remaining which are similar to them. 
1 We are not, for the moment, concerned as to whether this change can be made 
without travelling outside our data; some suggestions as to this are, however, made 
later. See p. 673. 
