20 MR. A. B. KEMPE OR THE THEORY OF MATHEMATICAL FORM, 
confine our attention to such constituents as will exhibit the form. Thus in the case 
of sec. 100 the constituent a C shows that the pair a, c is a symmetrical one. 
C Cb 
111. A discrete heap w T ill be represented by a table of one row ; a single heap of 
n units by a table of n rows; and a heap consisting of single independent heaps 
of p, q, r, s, . . . units respectively will have a table of \pX \qX |?’X [s . . . rows. 
Correspondences of Undistinguished Collections. 
112. The various correspondences of undistinguished collections are indicated by 
two-row constituents of the tabular representation of the system of which the 
collections are components. 
113. If the two rows of the constituent contain like letters in different orders, the 
constituent will be said to indicate a self-correspondence, i.e., a correspondence of a 
collection C with itself; and not with another collection (sec. 76). Any two-row 
constituent containing two complete rows of the table representing a system S will 
indicate a self-correspondence of S, and the table may accordingly be said to indicate 
the form of S by indicating all its self-correspondences. 
114. If the two rows of the constituent representing a correspondence of C with 
itself contain like letters in the same order, the constituent will be said to indicate 
an identical-correspondence of C. The identical-correspondence may be regarded as 
included among the self-correspondences of C. The self-correspondence of a single 
unit is an identical-correspondence. 
115. In general the table representing a system S will have rows which are 
partially alike, indicating that in some of the self-correspondences of S some of the 
components are identically-correspondent. For example, in sec. 100 the unit e is 
always identically-correspondent. Thus constituents may have several rows which 
are duplicates. If we merely desire to consider those correspondences which the 
constituent indicates, we may, of course, omit duplicate rows. 
116. If one two-row constituent be a part of another, the correspondence indicated 
by the former may be said to “ occur in " that indicated by the latter ; as when the 
latter correspondence occurs the former also occurs. 
117. If we regard all collections of units under consideration in an investigation as 
components ol the universal system which comprises all units, or, as is sufficient, as 
components of the whole system of units considered in the investigation, we may 
regard every correspondence of undistinguished collections as occurring in one or 
more self-correspondences of the universal or more limited system. From this point 
of view we see that a correspondence of two undistinguished collections, or the self¬ 
correspondence of a collection, restricts the possible correspondences and self-corre¬ 
spondences of other collections. In some cases it may determine absolutely the 
correspondences and self-correspondences of the other collections, not permitting 
