MR. A. B. KEMPE ON THE THEORY OP MATHEMATICAL FORM. 
65 
and a is accompanied by a class a such that 
a-\-a—u 
aa—z. 
Also we have 
(a + 6 + c+d+ . . .) — a bed... 
(cibcd . . . ) — ci d - b d - c -f- d d - • • < 
where a, b, c, d, . . . are any classes of the full set. Thus the relations of the com¬ 
ponent classes of a full set are similar to those of the component classes of a whole 
system. The class a may be called the “ obverse of a in the full set.” It is not of 
course the true obverse of a, i.e., is not a, unless the full set is the whole system. 
365. Any collection of n classes a, b, c, d, . . . which are such that they and their 
various sums and products are all different classes may be said to be a collection of 
unrestricted classes. The minimum full set of which a collection of n unrestricted 
classes is a component contains 2 2 “ classes; for the separated classes of such full set 
are abed . . . , abed . . . , a'b'cd . . . , abc’d . . . , a'b'c'd . . . , a'b'c'd' . . . , &c., 
which are 2 n in number. This full set may be said to be derived from the collection 
of n unrestricted classes. 
366. If a, b, c, cl, . . . be any collection of unrestricted classes, we can always find 
a class A of the derived full set, such that the classes \a, \b, Ac, A cl, . . . have any 
desired class relations to each other.'" 
367. Thus if we have a collection of 2 n unrestricted classes a, b, c, cl, ... , there 
will be a class A such that A a, A b, Ac, A cl, . . . constitute a full set, and another class p 
such that pa, p6, pc, \xd, . . . are a collection of separated classes. 
368. Suppose we have under consideration a full set T of 2’ 1 classes. We may, if 
we please, ignore the relations of inclusion, &c., which they have to each other, and 
treating them as a collection of separated classes constitute a derived full set P of 2~" 
classes containing them, their aggregates, and a zero class. Here the units of T are 
the products ra, rb, re, rcl, ... of a class r and a collection a, b, c, d, . . . of 2" un¬ 
restricted classes. When we ignore the relations of inclusion, &c., which ra, rb, tc, 
rcl, . . . have to each other, we no longer deal with ra, rb, tc, rd, . . . but with a 
collection na, rrb, i tc, ttcI, ... of separated classes. 
369. The obverse in T (sec. 364) of any class ra is to! , the class containing all 
classes of T which have only the zero class in common with ra. The obverse in 
P of 7ra is 7 tci , the class which contains all the separated classes irb, 7 tc, 7 tcI, . . . ; 
viz., all those classes arrived at by ignoring the relations of inclusion, &c., of rb, 
tc, Td, . . . 
* Thus Mr. Venn in his ‘ Symbolic Logic,’ at chap, v., employs intersecting ellipses to represent un¬ 
restricted classes. He then regards parts of these as eliminated, and so makes the ellipses represent 
classes having any desired relations. Here X represents the class composed of the uneliminated spaces. 
MDCCCLXXXVI. K 
