L—NOTATION AND TERMINOLOGY. 11 
gate of frequencies of the second order, and the twelve classes of 
the second order which can be formed where three attributes 
have been noted may be grouped into three such aggregates. 
12. Class-frequencies £ ay in tabulating, be arranged so that 
frequencies of the same order and frequencies belonging to the 
same aggregate are kept together. Thus the frequencies for the 
case of three attributes should be grouped as given below ; the 
whole number of observations denoted by the letter I being 
reckoned as a frequency of order zero, since no attributes are 
specified :— 
Order 0. WN 
Order 1. (4) (B) (7 
(a) (B) i 
Order 2. (4B) (40) « 
4p) (dy) 
(aB) (aC) | . () 
(a3) (a7) (, 
Order 3. (ABC) (a BC) 
(4By) (aBy) 
(ABC) (afC) 
(487) (apy) 
13. In such a complete table for the case of three attributes, 
twenty-seven distinct frequencies are given :—1 of order zero, 6 
of the first order, 12 of the second, and 8 of the third. It 
is, however, in no case necessary to give such a complete 
statement. 
The whole number of observations must clearly be equal to the 
number of 4’s together with the number of a’s, the number of 
4’s to the number of 4’s that are B together with the number of 
4’s that are not B ; and so on,—i.e. any class-frequency can always 
be expressed vn terms of class-frequencies of higher order. Thus— 
N=(4)+(a)=(B)+(B)=ete. 
= (LR) + (4B) + (aB) + (af3) = ete. @) 
(4)= (4B) + (48) = (40) + (47) =eto. | 
(4B) = (4BC) + (4 By) = ete. ) 
Hence, instead of enumerating all the frequencies as under (1), 
no more need be given, for the case of three attributes, than 
the eight frequencies of the third order. If four attributes had 
been noted it would be sufficient to give the sixteen frequencies of 
the fourth order. 
The classes specified by all the attributes noted in any case, 
t.e. classes of the nth order in the case of n attributes, may be