Karl Pearson 
297 
and thus 
Again 
V N 
N' 
N N 
where the summation S' is from x = x to cc ; thus 
1 
y V2tt 
''l I 2 
= ^ + r^yz. 
Accordingly we have 
and accordingly 
2^,2 (^2 _ 9 11 
xvn 
Now substitute (xvi) and (xvii) in (xiv) and we find 
(1 - -rff 
N (1 + Y 
W \z 
+ (1-7^2)2^. 
Or, 
( n,n, \zj N 
2 _ 
(1-^2)2 
N(l+ y2)2 L 
N 
' z ( ngUg 2., iV 
1 fyY 
]\f2 
n-,,n. 
(1 - v') 
r]'^ \ Z/' 
n,N J 
.(xviii). 
This form for the value of a^^ shows that the probable error of rj does not become 
infinite when 17 = 0 ; for in that case, we shall have — = ^4 for all values of s, 
n, N 
and — = ~ , so that the last two terms in l/rj^ take indeterminate forms which need 
Zj z 
evaluating. For calculation it is best to use the forin 
„ 2 „ (1 - v'f 
' N{1+ y2)2 
1 \^i^2fy\- 
W + yHv' + 
N 
+ 
Biometrika xi 
.(xix). 
20 
