356 
MR. R. A. FISHER ON THE MATHEMATICAL 
curve the departures from normality specified by the Pearsonian formula agree with 
those of that system of curves for which the method of moments gives the solution of 
maximum likelihood. 
The system of curves for which the method of moments is the best method of fitting 
may easily be deduced, for if the frequency in the range clx be 
y (x, 0i, 0 2 , 0b 0 4 ) dx, 
then 
must involve x only as polynomials up to the fourth degree ; consequently 
— g-a?(.x*+pix 3 +p2X 2 +p 3 x+iH) 
the convergence of the probability integral requiring that the coefficient of x l should be 
negative, and the five quantities a, p u p 2 , p 3 , p :l being connected by a single relation, 
representing the fact that the total probability is unity. 
Typically these curves are bimodal, and except in the neighbourhood of the normal 
point are of a very different character from the Pearsonian curves. Near this point, 
however, they may be shown to agree with the Pearsonian type ; for let 
y = Ce - a 
..2 T* 
-E5+*'5 + fcfi 
represent a curve of the quartic exponent, sufficiently near to the normal curve for the 
squares of /q and h, to be neglected, then 
A 
dx 
log y = -~. 
(T 
%(l-3ki- -idA 
W \ 
nr 
X 
(j 
cr 2 ( l + 3h- + 4frA 
(T 
neglecting powers of and k 2 . Since the only terms in the denominator constitute a 
quadratic in x, the curve satisfies the fundamental equation of the Pearsonian type of 
curves. In the neighbourhood of the normal point, therefore, the Pearsonian curves 
are equivalent to curves of the quartic exponent; it is to this that the efficiency of ; u : . 
and /x r in the neighbourhood of the normal curve, is to be ascribed. 
12. Discontinuous Distributions. 
The applications hitherto made of the optimum statistics have been problems in 
which the data are ungrouped, or at least in which the grouping intervals are so small 
as not to disturb the values of the derived statistics. By grouping, these continuous 
