68 
MR. A B. KEMPE OjST THE THEORY OF MATHEMATICAL FORM. 
represented by 2 m graphical units, of which pairs representing pairs of adjacent classes 
are linked, and pairs representing pairs of non-adjacent classes are not linked. 
381. A convenient mode of building up a graphical representation of such a system 
is the folio whig:—Suppose we have a graphical representation of any collection of 
units, say a diagram. To this we can add a precisely similar diagram, say we can 
repeat the diagram. The two diagrams will correspond unit to unit in one or more 
ways. Taking one of these ways only, we can link each graphical unit of one diagram 
to the corresponding unit of the other, say we can connect the two diagrams. Now 
draw a single graphical unit, repeat it, and connect the two diagrams (fig. 69). 
Repeat the resulting diagram and connect the two diagrams (fig. 70). Repeat the 
resulting diagram and connect the two diagrams (fig. 71). Repeat the resulting 
diagram and connect the two diagrams (fig. 72). This process can be carried on 
indefinitely, the number of units being each time doubled, so that the number of units 
after n processes is 2'\ 
382. Each of the diagrams may be said to represent a full set of the whole system 
finally arrived at. There are of course other full sets besides these in the system. 
Thus in fig. 72 there are 8 full sets represented, such as that in fig. 71 ; and generally 
in any system of 2 m units there are 2m full sets of 2 m ~ 1 units. 
383. We can pass from one graphical unit to another along links. We can do this 
by passing along a minimum number of links, or by more circuitous routes. The 
number of links passed over in the case in wdiich we pass over a minimum number 
may be termed the distance between the two units. Adjacent units are at a distance 
1 apart. 
384. In the following sections it is to be understood that when chains connecting 
two units are spoken of, those containing the same number of links as the two units 
are distant apart are referred to. We can, of course, by zigzagging about, make the 
chains much longer, but such chains are not those considered, which are only the 
chains of minimum length. 
385. Let cr 0 be any unit of a class system of 2 m units. There are m units linked 
to o-q, composing a collection which may be called <x l5 every unit of wdiich is at a 
771 (771 — 1 ) 
distance 1 from cr 0 . We have also a collection op of — —-— ; units other than cr 0 
and cr 1 , each of the units of which is at a distance 2 from cr 0 . So, generally, w r e have 
