MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
7 
without asserting whether the system be single or multiple. The result of this may 
of course be that we deal with different unirs at different stages of the investigation of 
S (secs. 126 ff). 
33. Systems are frequently more readily dealt with when regarded as components 
of more extensive systems. Much ingenuity has been expended on the discovery of 
systems, the addition of which to the particular system under consideration may assist 
in its investigation. It is to this that the existence of such units as substitutions, 
quarternions, quadrates, &c. (as to some of which I shall have to speak presently), is 
due. We shall see that we can always, by the addition of a proper system to any 
given system, completely define the latter by merely indicating the mode of distri¬ 
bution of certain pairs (secs. 81, 82). 
34. Notwithstanding the great assistance derived from the use of added systems, 
much reluctance is exhibited in employing them unless they can be shown to have 
their representatives in nature, i.e., unless “accidental” clothing can be found to fit 
them. The objections raised to symbolical methods which cannot be “interpreted” 
are strong' evidence of the fact that the accidental nature of much that comes under 
our consideration is not really appreciated. 
35. It is by no means unnecessary to state that the form of a system is independent 
of the particular method of defining it which we adopt; and that because it is easier to 
define a system by adding fresh units, it does not owe its form to the existence of 
those units. 
36. The system which is the actual Subject of investigation in an inquiry may be 
termed the base system ; those systems which are added for the purpose of assisting 
in the investigation being termed auxiliary systems. 
Heaps.—Graphical Representation of Units. 
37. There are two forms of systems of n units the consideration of which properly 
precedes that of all others. The one is that of a system which consists of n units, 
each of which is distinguished from each of the others, so that every component 5-ad 
is distinguished from every other component 5-ad for all values of s from 1 to n. A 
system of this form I shall term a discrete heap. 
38. The other is that of a system of n units which is such that every component 
5-ad is undistinguished from every other component s-ad for all values of s from 
1 to n. A system of this form I shall term a single heap. 
39. A discrete heap may be graphically represented by a number of small separated 
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Fig. 3. 
• Aft ’*4 • 
circles each containing a different letter (fig. 3). The letters render the graphical 
units distinguishable from each other. 
