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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