ERROR TO CASES OF NORMAL OlSTRIBUTION AND CORRELATfON. 119 
(vi.) the mean product of a {a — a) h c {y — y) . and 
«'(a' -a) + h' - ; 8 ) 4 - c' (/ - y) + . . . is 
+ bh'/B + cc'y 4- • • • ) ~ + cy 4" • • • ) («'a 4- h'^ + c'y 4* . . . )}/n. 
( 2 .) Let X;, denote the mean /;th power of a (a — a) 4 - b {/3' — /3) c {y — y) . 
The proportion of cases in which the numbers drawn from the different classes are 
p, (j, r , where q r = n, is 
\p + fl + r + 
Y . . 
Hence the mean ^’th power of aa' 4- bj^' 4- cy + • • • is 
11 . . . {ap 4 - bq 4 - cr 4 - . • 
w'-' \k X coefficient of SSS . . . ''' a^'By . . . ^ 
\p\q\p... ' 
— 11 
\k X CO. 6'' in (ae"^)L (ye^'’)' . . 
= k X CO. O’" in (ae"® "'' 4 " /Se** "’' 4 “ ye"®"* 4 ~ • • •) • 
Denote aa 4“ ^^8 4- cy + • • • by o). Then, since a' 4- , 8 ' -f- y' + . . . = 1 , 
a {a — a) 4- 6 — /3) 4- c (y' — y) 4- ... 
— ua 4" bfi 4~ t'y ”1“ • • • — ^ ip- ~1~ /8 y “!”•••) 
= {a — oj) a' 4" y8 + (c — fa>) y' 4" • • • 
Hence, writing a — co, b — oj, c — w, . . . for a, b, c, . . . , in the above result, we 
see that 
X^. = jZ: X coefficient of 0^' in 4- 4 - ygF-")®/'* 4 . ^ ^ ^j»-_ 
§16. Tendency of Distribution to become Normal .—We have now to prove that, 
when 71 becomes very great, the distribution of values of a (a — a) -j- b (j8' — /3) 
4- c (y — y) 4- . . . tends to become normal. To do this, we can use either the 
geometrical or the statistical definition of the normal curve. Of the two methods, 
the latter is the simpler. 
( 1 .) Since the mean square of a (a' — a) b {/S' — y 8 ) 4 " ^ (y' — y) + • • • varies 
Inversely as a, it is more convenient to find the distribution of 
v /n [a [a — a) -f b {/5' — -f c (y' — y) 4- . . .]. 
