6 
MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
23. Just as collections of units break up into single collections of undistinguished 
units, so collections of pairs break up into single collections of undistinguished pairs, 
and, generally, collections of n-ads into single collections of undistinguished n-ads. 
24. Differences between the various units, pairs, &c., of a collection may be termed 
the internal differences of that collection. Differences between the various units, 
pairs, &c., of a collection A and the units, pairs, &c., of a detached collection may be 
said to be external to A. 
Systems. 
25. If every component unit of a collection is distinguished from every unit which is 
detached from the collection, the collection will be termed a system. 
26. Every collection of units is a component of some system. 
27. The whole collection of units which come under consideration in any inquiry 
is a system ; for the units are distinguished from all others by being the only ones 
considered. 
28. The n-ads of one single system of units are distinguished from all those of any 
other single system of units, and from all the n-ads connecting any two systems of 
units. The connecting n-ads of any n systems of units are distinguished from those of 
any other n systems of units, and themselves break up into single systems of n-ads. 
29. The units of a single system of units must be dealt with as a whole ; for, as they 
are undistinguished from each other, no definition can be given of, or remark made 
about, one which is not equally applicable to each of the others. Each can only be 
spoken of as “one of” the units of the system, A similar observation applies in the 
case of single systems of pairs, triads, &c. 
30. Many systems of units are defined by stating that certain of their components 
constitute a system. It must be borne in mind that such a statement does not mean 
that the components are undistinguished from each other, but merely that they are 
distinguished from all others; the system of components may be a multiple one. It 
is by no means unnecessary to emphasize this, as we are prone to assume that the 
mode of classifying things is that of putting like things into the same class, rather than 
that of putting unlike things into different classes. 
31. The distribution of the various distinguished and undistinguished components 
of a system is regulated by definite laws ; so that a knowledge of the mode of 
distribution of some only of the distinguished and undistinguished components may 
determine the form of the system. There are in general several ways in which the 
form of a given system may be thus determined, and accordingly various different 
definitions of the same system may be adopted. 
32. For the statement of some of the properties of a system S it may be necessary 
to have the form fully defined ; in the case of others this may not be necessary, it being 
sufficient to state that certain components are distinguished, without asserting 
anything as to others, i.e., to state that certain components constitute a system, 
