MR. A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
67 
(0) a class Z, 
(1) m separated classes each containing Z, 
( 2 ) 
m(m— 1 ) 
o 
classes each containing two separated classes, 
y) -—classes each containing r separated classes, 
(m) a class U containing all the separated classes. 
Each class of the collection (r) may be derived by adding to some class s of the 
collection (?’—]) a class of (1) not contained in s. 
377. We have apparently as an essential accompaniment of the idea of classes the 
notion of the inclusion of one class in another. Inclusion is usually regarded as a 
relation between two classes. This is not, however, really so. The notion of inclusion 
essentially involves that of a chain of classes with two terminals, viz., a class U which 
contains all classes, and a class Z which is contained in all classes, of the chain. In 
dealing with a class A we really deal also with the classes U and Z ; and in dealing 
with two classes A and B we deal with the four classes A, B, U, Z. When this fact 
is recognised, inclusion, as commonly conceived, is seen to be a relation which, as 
far as the processes of exact thought are concerned, is accidental; that which is 
essential in it depending upon the places occupied in a system of classes by two 
classes relatively to two others (sec. 388). 
378. The classes U and Z are not, apart from accidents, distinguished from any 
other classes. The system of which the units are classes is a single one. The reason 
why U and Z seem to hold exceptional positions is that when we discuss classes we 
consider their positions with reference to U and Z, which, being constantly under 
consideration, acquire an accidental importance. 
379. There are two distinct sorts of inclusion, viz., direct and indirect. If a is 
contained in b, and b in c, then the inclusion of a in c may be said to be indirect ; if a 
is contained in c, and there is no class b under consideration such that a is contained 
m b and b in c, then the inclusion of a in c may be said to be direct. The two classes 
a and c may in the latter case be said to be adjacent, in the former non-adjacent. 
The relations of adjacency and non-adjacency are relations between tivo classes, and 
do not depend upon the consideration of the classes U and Z. In a system of 2 m 
classes every class is adjacent to m classes. Thus in the case of the collections of 
sec. 376 every class of the collection (r) is adjacent to r classes of the collection 
( r — 1) and m — r of the collection (r+l). 
380. The form of a system of 2 m classes is completely defined by the division of the 
pairs of classes into adjacent and non-adjacent pairs. It may therefore be graphically 
K 2 
