MR, A. B. KEMPB ON THE THEORY OF MATHEMATICAL FORM. 
69 
a collection ay of 
\r m—r 
units, each of which is at a distance r from <r 0 . 
When r—m 
we get a single unit oy which is the obverse of ay, and is unique with respect to cr 0 , 
being the only unit of the system which is so. The units of any collection 07 are 
each linked to m units of <r r -\ and to m—r of oy +1 , but to no other units of the 
system, so that none of them are linked to each other. We see here that the 
relation of a class cr 0 to its obverse c r m is one not depending on the relations of oy 
and c r m to U and Z, though expressible in terms of such relations. 
386. Consider now two collections of chains of links, (1) consisting of chains from 
ay to a unit a, (2) consisting of chains from oy to a unit (3 (see sec. 384). There are some 
graphical units through which chains of both collections pass. If we classify these 
according to their distances from oy, we find that the number in each class is always 
greater than 1, except in the case of the greatest distance, when there is one only, 
which is therefore unique with respect to cr 0 , a, (3. If the chain from oy to a passes 
through (3 , it is clear that the unit is (3 itself. In the same way, if we take any 
number of units a, (3, y, 8, . . . , there is a unit unique with respect to oy, a, (3, y, 8, 
which may be one of the units a, (3, y, 8, . . . We may obtain a like unit unique 
with respect to oy, a, (3, y, 8, . . . We may term the unit in the former case the 
product of a, /3, y, 8, . . . with respect to oy.; and in the latter case the product with 
respect to oy. 
387. If oy be Z, then the product of any units with reference to oy will be what 
has been spoken of in sec. 360, simply as the product of the classes represented by 
those units. The product with respect to oy, which will be U, will in such case be 
that which was defined in sec. 360 as the sum of the classes. 
388. If a chain from U to Z passes through two units a and (3, first through a and 
then through (3, then a and (3 represent classes which are such that a contains (3. 
If no chain from U to Z passes through both a and (3, then neither class is contained 
in the other. This makes it clear that the relation of inclusion is one in essentials 
depending merely on the form of a system of classes, and the position two units 
occupy in it relatively to two others (U and Z); or rather, as U and Z are unique 
with respect to each other, to one other, viz., either U or Z. 
389. Suppose we have a system S containing the n unrestricted classes a, b,c, d ,..., 
the obverses U and Z, and the collection C of separated classes abccl . . . , abed . . . , 
ab'c'd . . . , &c. Here C has the same self-correspondences as a single heap. Now, if we 
confine our attention to those self-correspondences of S in which the collection of 2 n 
units consisting of a, b, c,d,..., and their obverses is self-correspondent, the collection 
C will have those self-correspondences only which are characteristic of a full set; and 
will not have any correspondences with other collections; also the other units of S 
(except Z) which may then be regarded as aggregates of the units of C taken two, 
three, &c., together will correspond only if they are aggregates of component collec¬ 
tions of C which correspond ; i.e., they and C will have correspondences as if they 
