MR. A. B. KBMPE ON THE THEORY OF MATHEMATICAL FORM. 
a 
one state of things exists, of one form, and at another time t' of another form. This 
is, however, not so ; the form of a collection is absolutely invariable. The apparent 
alteration of form arises from overlooking the fact that the units dealt with are not 
x, y, z, &c., but (x at time t), (y at time t), &c., (x at time t'), (y at time t'), &c. The 
unit (x at time t) may be quite different from ( x at time t'). Where there is only one 
alternative t, the collection (x at time t), (y at time t ), (z at time t), &c., is of the same 
form as the collection x, y, z, &c., and it matters not which is the collection dealt with. 
16. Aggregations of those things which, in a symbolical representation of the units 
considered in any case, are already represented as units, must not be supposed to be 
sufficiently represented without additional symbols, but must each be represented by 
a distinct symbolical unit. An aggregation of things is, as far as the processes of 
reasoning are concerned, a mere unit, and must be so dealt with. 
17. While it is important that a unit should be represented and dealt with as a 
unit, it is equally important that we should not be misled by our modes of thought 
and consequent use of language into regarding a number of distinct units as one only. 
A collection of units must in a symbolical representation be represented as a single 
unit where it is so regarded ; but where the word “ collection ” is used, as it is here 
(sec. 8), merely to “ denote ” or mark off a number of things each of which is 
considered as a distinct unit, we must'be careful to represent each of those things by 
a distinct symbol. 
Some Definitions. 
18. Any collection of units which consists entirely of units selected from another 
collection will be termed a component of the latter. Any aspect of a component of a 
collection may also be spoken of as a component of the collection. 
19. Two collections of units will be said to be detached if they have no component 
in common. 
20. An n- ad which has one or more units in common w T ith each of a number of 
collections will be said to be an n- ad connecting those collections ; e.g., the pairs which 
a single unit makes with the various units of a collection will be termed the pairs 
connecting the unit and the collection. An n-ad connecting n detached collections has 
one unit and one only in common with each. If A, B, C, D, .... be collections 
of which a, h, c, d, ... are component units respectively; when an n- ad connecting 
A B C D . . . is spoken of, it must be understood that an ?i-ad such as a b c cl ... is 
meant, and not one such as h d c a . . ., which will be spoken of as connecting 
B D C A . . . 
21. If the component units of a collection are all undistinguished from each other 
the collection will be said to be single. 
22. Units which are undistinguished from the same unit are undistinguished from 
each other ; thus if a collection of units is not single it consists of two or more 
detached single collections, and will be termed a double, treble, &c., collection, accord¬ 
ing to the number of component single collections which it contains. 
