THEQRY OF STATISTICS. 
instead of the simpler symbols & (4) (B) (4B). Similarly, the 
general relations (2), § 13, Chap. 1, using U to denote the common 
attributes of all the members of the universe and (I) conscquently 
the total number of observations 4, should in strictness be written 
in the form— 
(U) =(UA)+(Ua)=(UB)+ (UB) =c¢te. 
= (UAB) + (UAB) + (UaB) + (Uap) = ete. 
UA) =(UAB)+(UAB)= (UAC) + (Udy)=ete. 
UAB) =(UABC) + (UABy) = ete. 
3. Clearly, however, we might have used any other symbol 
instead of U to denote the attributes common to all the members 
of the universe, e.g. 4 or B or AB or ABC, writing in the latter 
case— 
(ABC) = (ABCD) + (4BCY) 
and so on. Hence any attribute or combination of attributes 
common to all the class-symbols in an equation may be regarded as 
specifying the universe within which the equation holds good. 
Thus the equation just written may be read in words: The 
number of objects or individuals in the universe ABC is equal to 
the number of D’s together with the number of not-D’s within 
the same universe.” The equation 
(AC) =(4BC) + (480) 
may be read : ‘The number of 4’s is equal to the number of 4’s 
that are B together with the number of 4’s that are not-B 
within the universe C.” 
4. The more complex may be derived from the simpler relations 
between class-frequencies very readily by the process of specifying 
the universe. Thus starting from the simple equation 
(a) == (4), 
we have, by specifying the universe as (3, 
(B)= (8) - (48) 
=N-(4)-(B)+ (4D). 
Specifying the universe, again, as y, we have 
(aBy) = (7) - (Ay) = (By) + (4.By) 
=N-(4)-(B)—(C)+ (4D) + (4C) + (BC) - (4BC0). 
5. Any class-frequencies which have been or might have been 
observed within one and the same universe may be said to be 
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