32 MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
F is that of a single heap H of m units. Here F' being the form of any system S, the 
compound system S H differs from S in that we have in lieu of any unit a of S a single 
heap of m units, each unit of which may be called a, the form relations between the 
connecting r-ads of r of the heaps being the same as those existing between the r-ads 
of S. Thus if in S we have abcd^> - <ipgrs, in S H we shall also have cibcdp> -< 
pqrs, whatever units of the heaps aaa . . ., bbb . . . , &c., we select. 
192. If we combine two connected component collections L and M of S, the result¬ 
ing- collection cannot contain those units which are common to L and M twice over; 
i.e., the result of combining the collections a, b, c, d, and c, d, e, f is the collection 
a, b, c, d, e, f. If in S H we select collections a, b, c, d and c, d, e, f so that the units 
c, d of the first are not the same units as c, d of the second, the sum of the two 
collections will be the collection a, b, c, c, d, d, e, f. 
193. The third mode of composition is that in which the derived system has all the 
self-correspondences which each of a number of systems, of the same number of units, 
has. We may represent the units of such a system by symbols 
(Aa . . . ), (B6 . . . ), (Cc . . . ), &c., 
where A, B, C . . . are units of one of the compounded systems, a, b, c those of 
another, and so on. 
General Method of Graphically Representing a System. 
194. Let each unit of a system S, each unified column, each unified row, and 
each element of its tabular representation, be represented by a graphical unit, using 
different kinds of graphical units in the case of the units of S, the rows, columns, and 
elements respectively, four kinds in all. Connect each graphical unit which represents 
an element by links to 
(1.) the graphical unit representing the unit of S of which the element represents a 
unit aspect; 
(2.) the graphical unit representing the unified row in which the element lies ; 
(3.) the graphical unit representing the unified column in which the element lies. 
We get a graphical representation of a system of 2n-\-m-\-mn units, of which S is 
a component system. 
195. It is obviously not always necessary to employ the somewhat cumbrous mode 
of graphical representation here given; simpler methods can be adopted in special 
cases. Frequently, as we have already seen, it will not be necessary to have any 
auxiliary graphical units, but merely those representing the units of S itself. 
Networks. 
196. The pairs of any single collection of pairs are either all component pairs of a 
single system of units, or all connecting pairs of two single systems of units. 
