Miscellanea 
211 
hypothesis I found that the slightest correlation produces a very marked retardation in the 
approach to normality with increase in the size of the sample. 
It will be observed that this is essentially a 'small sample' problem, for with increase in the 
size of the sample the correlation due to likeness between successive individuals diminishes 
except in exceptional cases when it becomes manifest as a well-marked heterogeneity. 
My results emphasize the necessity of avoiding anything which tends to produce secular 
variation and as far as possible to neutralise it by repeating observations only after some time 
has elapsed. 
Thus repetitions of analyses in a technical laboratory should never follow one another but an 
interval of at least a day should occur between them. Otherwise a spurious accuracy will lie 
obtained which greatly reduces the value of the analyses. 
In the present case there is not sufficient evidence to show whether correlation was really 
present, but as in the course of a fairly extended practice I have not yet met with observations 
in which this tendency was altogether absent, I incline to the belief that it was. 
In any case, being ignorant of the technique, I can only suggest as possibilities slight varia- 
tions from point to point on the slide, differences in light or in the observer as the day went on. 
The general problem is as follows : 
Let samples of n be drawn from a population with constants /ij, fi4, iSi, ^2, ^"■nd let the 
samples be drawn in such a manner that the individuals composing each sample are corre- 
lated with correlation coefficient r, then, assuming linear regression and hornoscedastic arrays, 
the constants of the distribution of their means {M., , J/3 , iI/4 , Bi , B-i) are as follows : 
J/3=g {1 -f (n - 1) r) {1 + {2n - 1) r}, 
M,= [f4 {1 + (3^1 - 1) r + Sn (n - 1) f^} + 3 - 1)(1 - r){l+nr) 
_A {l+{2n-l)rY 
{1+ {3n-l)r + S7i (w-l)r^} 3 ( » - 1) (1 - r)(l 
71 (l + 2r){\ + {n-l)r} 71 {I +2r) {I +{n -I) r\' 
As the method of determining the three moment coefficients is the same in each case and it 
is merely a question of reduction to obtain Bi and Bo, it will be sufficient for me to give the 
proof for Mi. 
Let XiX-i...x„ be the values, measured from the mean of the population, of the individuals 
composing the typical sample, and let there be JV such samples. 
Then J/,= ^s|^l±fi±i:^'j* 
_ ^ S{x,i) + iS (xi ^x-i) + (.Ti^.r./) -I- 12,S' {.v{'.V2X3) + 24.S {x^x-jX^X j) 
~jy^ ' ' 71* ^ ^" 
Taking each of these six terms in turn we have 
S{.S'(.r.y,_ WS(Hx,l-i'' )_M4 
iV'ft* " JV.n* 7i' ^ 
For (S" (,-!;/) has 71 terms and when they are taken over all the N samples which compose the 
population there will be 71 . iix^ of Xi*, 7ixi being the number of Xi'a in the population and Wa-i.r., the 
number of .t-j's associated with x^s, and so on. 
27—2 
