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 18