OF ERROES OF JUDGMENT AND ON THE PERSONAL EQUATION. 
275 
(y.) That the contributions towards any individual error of these cause-groups are 
not correlated among themselves. 
It is not my purpose at present to consider the philosophical arguments for or 
against these axioms. I have considered the matter at length in a paper not yet 
published, but I want to indicate the source of such relations as those just referred to. 
If actual distributions of error do not sensibly satisfy pg = 0, then axiom (a) is not 
true; if they do not sensibly satisfy then either (j3) or (y) or both are 
invalid. 
The nature of the relation deserves a little fuller consideration from 
the physical side. 
Let ?n^ and be the number of errors of magnitudes and respectively, and 
let nix .2 and w'g/ represent the second and fourth moments of the remainder of the 
errors. Let -f- = m'. Then 
n/ji^ = -b m^x^-, 
nix^ — -j- 
Hence we find :— 
-h m'x^ + — n ix^ — m'x.^) {x^ -f x^). 
Now without altering the total frequency, i.e., keeping m' constant, take part of 
the frequency and transfer it from x^ to ; do this equally on both sides of the 
mean, so that the position of the mean be not altered. Now in order that /rg should 
also not be altered, x^, being supposed constant, we must have :—• 
Swq/m^ = {2x^Sx-^)/(x 2^ — x^^), 
or if x^ be > x^, must be positive. Thus if we bring a part of the outlying 
frequency inward to a point nearer the mean, we can still retain the same mean and 
the same standard deviation, i.e., get the same normal curve, if we shift the inlying 
frequency group a little outward. The whole effect of such a change will be to 
flatten the frequency curve at its summit by a reduction of its tails, which increases 
the middle part of the curve. Now, looking at the above value of we see that 
since < x^^, nfi^ — n'g/ — mx.^ is negative, and therefore that when x^ increases 
/X 4 , decreases. 
We conclude accordingly that symmetrical or nearly symmetrical curves which 
have the same mean and standard deviation as a normal curve will be flatter topped 
if be < 3/x/, and steeper at the top if /x^, be > S/x/. 
Take, for example, the details of shots at a target given by Merriman, ‘ Method 
of Least Squares,’ p. 14 : here* 
/X 2 = 2-402,343 , /x^. = 14-578,491 , 
and 3 /X 32 _ 17-313.752. 
* Using Sheppard’s corrective terms, ‘ London Math. Soc. Proc.,’ vol. 29, p. 369. 
2 N 2 
