122 
MR. W. F. SHEPPARD ON THE APPLICATION OF THE THEORY OF 
Now let n become indetinitely great, the point 0 remaining fixed. Then this 
relation holds all along the curve which is the limit of the polygon PqPi . . . P„, and 
therefore this curve is a normal curve of parameter 2\/a/3, having its central ordinate 
at O. The mean value of a is found by putting OM = 0, which gives a! = p^n = a. 
Thus the values of \/n (a — a) are distributed normaily with mean square 
= a (l — a) about a mean value zero; and therefore the values of a' — a are 
distributed normally with mean square a (1 — a)/n. 
(ii.) Next, consider the distribution of values of a — a when certain other errors, 
as y8' — and y' — y, have particular values. This distribution is found by taking 
an indefinitely great number of random selections, each containing n individuals, and 
isolating those sets in which the numbers drawn from the classes B and C are respec¬ 
tively and ny'. From the principles of random selection it follows that the distri¬ 
bution of values of a' — a in these sets is the s.uiip as if we made random selections of 
n {1 — yS' — y) individuals from that portion of the community which does not involve 
B and C. Of this portion of the community, the class A forms a part denoted by the 
fraction a/(l — /3 — y). Hence the values of no.', the number coming from A, are 
distributed with mean square n {1 — — y) X «(! — a — ^ — 7)/( ^ ^ — y)' 
about a mean value 5? (1 — /S' — y) X «/(! — ,5 — y). So long as yS' — yS and y — y 
are small in comparison v/ith yS and y, this is equivalent to sa 3 dng that the values of a 
are distributed with constant mean square about a mean value a {I — yS' — y')/(] —/3 — y) 
= a — X (yS' — yS) — X (y' — y), where X= a/(I —/3—y). Thus the distributions of a — a, 
yS' — yS, y' — y, . . . are normally correlated ; and therefore, since the separate distri¬ 
butions are normal, the values of a (a' — a )-f6(y8'-/8)-Fc(y'-y)+.. . are normally 
distributed. 
Since this argument only applies when a' — a, — yS, y — y, . • • are small, the 
result is subject to the limitation pointed out in (I) (above). 
§ 17. Frohahle Error and Probable Discrepancy .—Let X be any magnitude which 
is determined by observation of the ratios a', y8', y', . . . Then X can be written in 
the form / (A, y8', y', . . .). Now suppose n to be very great. Then the values of 
a — CL, yS' — y8, y — y, . . . are distributed normally with mean values zero and mean 
squares a(l — oL)/n, y8(l —/S)/n, y (I — y)Ab • • • ; and therefore it may be supposed 
that in any particular case the values of a! — a, y8' — /S, y — y, ... will be very 
Q„N„+i have the same slope at the points PjPa • . . Pa+i as the respective curves 2 which pass through 
those points. 'Professor Karl Pearson has arihved at a different result (‘Phil. Trans.,’A, vol. 186 
(1895) p. 357) by forming the polygon P.P. . . . P,+i and finding the “ slope ” at the middle points of its 
sides. There is of course no discrepancy between the two I’esults, since they deal with different polygons, 
and with points having different relative positions on these polygons. The curve found by Professor 
Pearson becomes the normal curve when n is made indefinitely great. 
To prevent misunderstanding, it should be pointed out that, in either case, the slope of the polygon 
at the points in question is not the same as the slope of any one curve of the family considered. 
Professor Pearson’s statement (op. cit., p. 356) as to the existence of a close relation between the 
binomial polygon (for = /i) and “ the ” normal curve seems to require some qualification, 
