MR. A. B. KEMPE OR THE THEORY OF MATHEMATICAL FORM. 
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126. Many of the relations which collections of units hold to each other are 
apparently independent of the form of the collections, or of the form of the 
system S, of which they are components. In dealing with such relations we ignore 
differences, and regard the units composing the collections as single heaps, i.e., we do 
not deal with the units a, b, c, d, . . . of S, but with others a, ft, y, S, . . . which con¬ 
stitute a single heap, but have the same self-correspondences as a, b, c, d, . . . have as 
long as certain other units remain identically-correspondent. 
127. Since we can in all cases ignore differences, any system S may be regarded as 
a single heap, the peculiarities of form which it possesses in any particular investiga¬ 
tion being regarded as due to the fact that we are not considering S alone, but in 
conjunction with other units, those correspondences only of S being dealt with which 
admit of the additional units remaining identically-correspondent. All statements, 
therefore, as to the distinguishableness and undistinguishableness of components of S, 
and as to their being of particular forms may be taken as relative, viz., as statements 
that the components have the correspondences characteristic of those forms as long as 
certain units detached from S remain identically-correspondent. 
128. It may be laid down generally, that in almost every instance where we seem 
to investigate a base system S which may be regarded and spoken of as being of 
n units and of a particular form, we really deal with a single heap system H of 
n units and a system F which remains identically-correspondent while the units of H 
go through the correspondences characteristic of the system S. 
129. Any collection of units which while another collection C remains identically- 
correspondent has self-correspondences characteristic of a collection of a special form 
may be said to be of that form relatively to 0. 
Sets. 
130. If abed . . . , pqrs . . . , are undistinguished components of a collection 
a, b, c, d, . . . p, q, r, s, . . . I, m, n, o, ... , then the units tv, x, y, z, . . . which are 
such that abed . . . Imno . . . >-< pqrs . . . ivxyz . . . may or may not be units of 
the collection, and in some cases cannot be selected so as to be units of the collection. 
If the collection be such that whatever undistinguished components abed . . . , pqrs . . . 
we select, and whatever other component Imno . . . we select, w, x, y, z ... can 
always be selected from the collection, then the collection will be termed a set. 
131. A system is obviously a set. A set is not necessarily a system ; it may be 
one of a number of undistinguished sets wdiich together compose a single or multiple 
system. In most investigations our inquiries are directed towards the discovery of 
the forms of component sets of the base system. As far as the distribution of its 
distinguished and undistinguished components is concerned, a set in no way differs 
from a system. 
132. In the self-correspondences of a set every correspondence of its undistinguished 
