te THEORY OF STATISTICS. 
Arithmetical work, however, should be executed from first 
principles, and not by quoting formule like the above. 
Example ii.—Check the work of Example i., § 13, by finding the 
frequencies of the ultimate classes from the frequencies of the 
positive classes. 
i = (4B) - (4BC)=338 —57=281 
Ay) = (dy) - (ABy)= (4) - (40) ~ (4By) 
=877 - 143 — 281 =453 
(aBy) = (By) - (4By) =H = (B) - (C) + (BO) - (43) 
=10,000 — 1086 — 286 +135 — 453 
=10,135 — 1825 = 8310 
and so on. 
18. Examples of statistics of precisely the kind now under 
consideration are afforded by the census returns, e.g., of 1891 or 
1901, for England and Wales, of persons suffering from different 
“infirmities,” any individual who is deaf and dumb, blind or 
mentally deranged (lunatic, imbecile, or idiot) being required to 
be returned as such on the schedule. The classes chosen for 
tabulation are, however, neither the positive nor the ultimate 
classes, but the following (neglecting minor distinctions amongst 
the mentally deranged and the returns of persons who are deaf 
but not dumb) :—Dumb, blind, mentally deranged ; dumb and 
blind but not deranged; dumb and deranged but not blind; 
blind and deranged but not dumb ; blind, dumb, and deranged. 
If, in the symbolic notation, deaf-mutism be denoted by 4, blind- 
ness by B, and mental derangement by C, the class-frequencies 
thus given are (4), (B), (C), (4By), (480), (aBC), (ABC) (cf. 
Census of England and Wales, 1891, vol. iii., tables 15 and 16, 
p. vii. Census of 1901, Summary Tables, table xlix.). This set of 
frequencies does not appear to possess any special advantages. 
19. The symbols of our notation are, it should be remarked, 
used in an inclusive sense, the symbol 4, for example, signifying 
an object or individual possessing the attribute 4 with or without 
others. This seems to be the only natural use of the symbol, 
but at least one notation has been constructed on an exclusive 
basis (cf. ref. b), the symbol 4 denoting that the object or in- 
dividual possesses the attribute 4, but not B or C or D, or what- 
ever other attributes have been noted. An exclusive notation is 
apt to be relatively cumbrous and also ambiguous, for the reader 
cannot know what attributes a given symbol excludes until he 
has seen the whole list of attributes of which note has been 
taken, and this list he must bear in mind. The statement that 
the symbol A is used exclusively cannot mean, obviously, that the 
object referred to possesses only the attribute 4 and no others 
yd