114 
On the Probable Errors of Frequencn Constants 
a first approximation, i.e. if we may neglect the terms in as compared with those 
. 1 
m — : 
ViV 
yMi = ^n, (1). 
Similarly for a second ordinate 
y^hxi = 8)1.2 (1 bis). 
Such equations connect the change in the grade with the change in the frequency 
and provide at once the relations 
0".ri = 0"«i/2/l. 0-Xo = 0-nJlh (2). 
If it be suggested that we shall have for a given sample a whole series of very 
irregular distributions of the individuals beyond y^, the reply is that we are 
seeking to find for all samples, and we are justified to our degree of approximation 
in considering that S.ri is the average grade change for all distributions which give 
an excess Shj beyond yi, and that such average grade change may be looked upon as 
resultinof from an area change and not from a scheme of isolated individuals. 
Now if N be large as compared to 31 we have at once 
1 - 
.(3),. 
which lead to 
1 
M 
., _ N tiu ! 
To our order of approximation [ terms in 
N 7^ 
y}N 
Nho 
y}N 
1 
71 
1- 
1 - 
.(4). 
N) 
we write finally 
ViV \ yJ 
1 - 
N. 
(f) 
Again 
Mean {h.i\hx.^ — 
1 
" ^N\y; 
Mean {hn^8m). 
1 - 
.(5). 
Now if we agree that x., shall be less than x-^, 
Hn = Hi + «2 — ''-i = ""i + «:) say. 
Mean (ShiS»,) Mean (Sthf + Mean {S>hS»,). 
Now 111 and are frequencies having no part of their ranges in common and 
accordingly if be large as C(mipared with M, 
Mean (SnMi) = - N 
with the same approximation as before. 
M M 
•(6), 
