L—NOTATION AND TERMINOLOGY. 13 
The ultimate frequencies form one natural set in terms of which 
the data are completely given, but any other set containing the 
same number of algebraically independent frequencies, viz. 27 
may be chosen instead. 
15. The positive class-frequencies, including under this head the 
total number of observations &, form one such set. They are alge- 
braically independent ; no one positive class-frequency can be ex- 
pressed wholly in terms of the others. Their number is, moreover, 
2", as may be readily seen from the fact that if the Greek letters 
are struck out of the symbols for the ultimate classes, they become 
the symbols for the positive classes, with the exception of afy 
. +. . for which # must be substituted. Otherwise the number 
is made up as follows :— 
Order 0. (The whole number of observations) . : 1 
Order 1. (The number of attributes noted) . : n 
Order 2. (The number of combinations of n things 2 together) ph 
Order 3. (The number of combinations of n things 3 together) aol) Fed 
and so on. But the series 
n(n—-1) n(n-1)(n-2) 
l+n+ 1.9 {55 Th eieite 
is the binomial expansion of (141) or 2", therefore the total 
number of positive classes is 2". 
16. The set of positive class-frequencies is a most convenient 
one for both theoretical and practical purposes. 
Compare, for instance, the two forms of statement, in terms of 
the ultimate and the positive classes respectively, as given in 
Example i,, § 13. The latter gives directly the whole number of 
observations and the totals of 4’s, B’s, and (’s. The former gives 
none of these fundamentally important figures without the perfor- 
mance of more or less lengthy additions. Further, the latter gives 
the second-order frequencies (4B), (4C), and (BC), which are neces- 
sary for discussing the relations subsisting between 4, B, and C, but 
are only indirectly given by the frequencies of the ultimate classes. 
17. The expression of any class-frequency in terms of the 
positive frequencies is most easily obtained by a process of step- 
by-step substitution ; thus— 
(@B) =(a)- (aB) 
=N-(4)-(B)+(4B) . ?) 
(afy) = (ap) - (aBC) 
=N - “4 - (B) + (4B) - (aC) + (a BC) 
=X (4) -(B)-(C) + (4B) + (AC) + (BC) = (4BC) (4) 
(c