K, Pearson 
271 
71 — 1 moments of the two ciu'ves, the higher moments of the curves become ipso 
facto more and more nearly identical the larger ?i may be. But such a term as 
vanishes if the higher moments are equal ; for we may write 
R=-<l>n (0) + ±^ (0) + . . . , 
\n + 1 ^ 
and accordingly 
r dR , c?0»(O) I , , 
+ d«r " i«T2 ^"^^"^^ " ^ ^ 
+ etc. 
Thus if A=A', we have the factors - (/tA„+i t^-'n+'i), etc. 
Thus besides the smallness of the factors ^-r" — , \ ^' , ...etc., depending on 
\n \n + \ 
the hypothesis of ctmvergency in Maclaurin's expansion, we have the smallness 
of the factors /jLn — i^n, P'n+i ~ fJ''n+i, ■■■ depending on the fact that if n—\ 
moments of a curve are equal, the succeeding ones will be nearly e(|ual. 
We conclude accordingly that equality of moments gives a good method of 
fitting curves to observations. It is this method of moments which I ventiii-e 
to suggest as a good systematic process, preferable to those in ordinary use 
when the method of least squares is too laborious or impracticable, for deter- 
mining the constants of empirical or theoretical curves from observation. This 
is really the method which has been long in constant use for fitting the normal 
curve of errors y = yoe^^' "^''^ to observations ; it has been largely adoptetl by myself 
in fitting skew frequency curves to observations* ; and it becomes identical with 
that of least squares when we fit parabolic curves of any order to observations. 
It is then no approximation, but the accurate solution, for the expansion by 
Maclaurin's Theorem is finite. 
One great advantage of the method, as will be illustrated below, is that it 
enables us to determine in many cases the whole of the theoretical curve from 
a part, if the observations can only be made along a portion of the range. 
There are three essentials to the application of the method : 
(a) We must be able to ascertain the moments of the theoretical curve in 
terms of Cj, c^, c^, ... Cn- 
* Phil. Trans., Vol. 186, A, pp. 34.S— 414. 
