THEORY OF STATISTICS. 
expressed symbolically by expanding the ultimate in terms of 
the positive frequencies, and writing each such expansion not 
less than zero. We will consider the cases of one, two, and 
three attributes in turn. 
8. If only one attribute be noted, say 4, the positive frequencies 
are V and (4). The ultimate frequencies are (4) and (a), where 
(a) = NN = (4). 
The conditions of consistence are therefore simply 
M40  N-(4)40 
or, more conveniently expressed, 
(@ (A)<0 (5) (A): AE. (1) 
These conditions are obvious: the number of 4’s cannot be less 
than zero, nor exceed the whole number of observations. 
9. If two attributes be noted there are four ultimate frequencies 
(4B), (4B), (aB), (eB). The following conditions are given by 
expanding each in terms of the frequencies of positive classes— 
(a) (4B)<0 or (45) would be negative 
(6) (AB) (4)+(B)-N ,, (af) ” ” (2) 
(c) (AB)}(4) » (45) ” ” ( 
(d) (4B)3(B) » (eB) ” ) 
(a), (c), and (d) are obvious; (b) is perhaps a little less obvious, 
and is occasionally forgotten. It is, however, of precisely the 
same type as the other three. None of these conditions are 
really of a new form, but may be derived at once from (1) (a) and 
(1) (6) by specifying the universe as B or as f respectively. The 
conditions (2) are therefore really covered by (1). 
10. But a further point arises as regards such a system of 
limits as is given by (2). The conditions (a) and (b) give lower or 
minor limits to the value of (4B); (¢) and (d) give upper or 
major limits. If either major limit be less than either minor limit 
the conditions are impossible, and it is necessary to see whether 
(4) and (B) can take such values that this may be the case. 
Expressing the condition that the major limits must be not less 
than the minor, we have— 
(4)40 { (B)<0 } 
4)» (B)»N 
These are simply the conditions of the form (1). If, therefore, 
(4) and (B) fulfil the conditions (1), the conditions (2) must be 
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