MR. R. A. FISHER OX THE MATHEMATICAL 
3?. 6 
so that, approximately, 
= -(p'+lp) 
n 
for large values of p, the efficiency of the method of moments is, therefore, approximately 
jsL+.±p 
p + l_p + 2p + 6 
Efficiencies of over 80 per cent, occur when p exceeds 38-1 (ft x — 0-102) ; evidently 
tlie method of moments is effective for determining the form of the curve only when it 
is relatively close to the normal form. For small values of p. the above approximation 
for the efficiency is not adequate. The true values can easily be obtained from the 
recently published tables of the Trigamma* function (11). The following values are 
obtained for the integral values of p from 0 to 5. 
V 
Efficiency 
0 1 2 3 4 5 
0 0-0274 0-0871 0-1532 0-2159 0-2727 
An interesting point which may be resolved at this stage of the enquiry is to find 
the variance of m, when a and p are not known, derived from the above set of simul¬ 
taneous equations ; that is to say, to calculate the accuracy with which the limiting 
point of the curve is determined ; such determinations are often stated as the result 
of fitting curves of limited range, but their probable errors are seldom, if ever, evaluated. 
To obtain the greatest possible accuracy with which such a point can be determined 
we must divide the minor of hJd } namely, 
cm " 
by 
whence 
AT-t d 2 i , ,v ,1 
«d p+1 rfy log(p!) “ 1 j 
g. _lj 2 y. i 0 g ( p t) _ i + 1 
a p — If dp p p\ 
d 2 
a* = ~ 
n 
a.y -qp+ljffikg (*>!)-! 
2 A lo 8' h :>~ f + h 
dp p p 
The position oFthe limiting point will, when p is at all large, evidently be determined 
with much less accuracy than is the position, as a whole, of a curve of known form and 
size. Let n’ be a multiplier such that the position of the extremity of a curve calculated 
It is sometimes convenient to write f (x) for ~ log (a-!). 
