4 
MR, A. B. KEMPE ON THE THEORY OF MATHEMATICAL FORM. 
the same number of units but having different distributions will be of different 
forms. Thus the two tetrads a, b, c, d, and p, q, r, s, of fig. 2, contain the same 
number (four) of unit spots, but they are of different forms ; for a, b, c, d are all 
undistinguished from each other ; while q, r, s, though undistinguished from each 
other, are all distinguished from p. The distribution of the distinguished and 
undistinguished pairs and triads is also obviously different in the two cases. The 
word “ form ” will in this memoir be always employed in the sense here indicated. 
TO. Two collections of units which are undistinguished are of the same form, but 
two collections which are of the same form are not necessarily undistinguished; there 
may be the same distribution of distinguished and undistinguished units, pairs, &c., 
in each, while the units, pairs, &c., of one are all distinguished from the units, pairs, &c., 
of the other. 
11. Each of the forms which a system of any number n of units can assume, owing 
to varieties of distribution, is one of a definite number of possible forms, and the 
peculiarities and properties of the collection depend, as far as the processes of reasoning 
are concerned, upon the particular form it assumes, and are independent of the dress, 
geometrical, algebraical, logical, &c., in which it is presented; so that two systems 
which are of the same form have precisely the same properties, although the garbs 
in which they are severally clothed may, by their dissimilarity, lead us to place the 
systems under very different categories, and even to regard them as belonging to 
“ different branches of science.” 
12. It may seem in some cases that other considerations are involved besides 
“ form,” but it will be found on investigation that the introduction of such con¬ 
siderations involves also the introduction of fresh units, and then we have merely 
to consider the form of the enlarged collection. 
13. In order to put form in evidence some ‘‘accidental” clothing is of course 
necessary ; iff however, we employ more than one species of clothing, each species 
being uniform and suited to forms of every kind, the likelihood of its accidental 
nature being overlooked will be reduced to a minimum. 
Units. 
14. The units which we have to consider exhibit endless variety; thus we may 
have a material object dealt with as one unit, a quality it possesses as another, a 
statement about it as a third, and a position it occupies in space as a fourth. The 
task of specifying the units which are considered in an investigation may in some 
cases be one of considerable difficulty, and mistakes are likely to occur unless the 
operation is conducted with great care. 
15. We have frequently to deal with things x, y, z, &c., pairs, &c., of those things, 
the differences between which depend on the existence or non-existence of certain 
circumstances, or upon taking into account or ignoring certain circumstances. Thus 
we may apparently have collections of units x, y, z, &c., which are at one time t, when 
