72 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
approaches with a close degree of approximation to the well-known error or probability- 
curve. A frequency-cniwe, which, for practical purposes, can be represented by the 
error curve, will for the remainder of this paper be termed a normal curve. When a 
series of measurements gives rise to a normal curve, we may probably assume 
something approaching a stable condition; there is production and destruction 
impartially round the mean. In the case of certain biological, sociological, and 
economic measurements there is, however, a well-marked deviation from this 
normal shape, and it becomes important to determine the direction and amount of 
such deviation. The asymmetry may arise from the fact that the units grouped 
together in the measured material are not really homogeneous. It may happen that 
we have a mixture of 2, 3, . . . n homogeneous groups, each of which deviates about 
its own mean symmetrically and in a manner represented with sufficient accuracy by 
the normeJ curve. Thus an abnormal frequency-curve may be really built up of normal 
curves having parallel but not necessarily coincident axes and different parameters. 
Even where the material is really homogeneous, but gives an abnormal frequency-curve 
the amount and direction of the abnormality will be indicated if this frequency-curve 
can be split up into normal curves. The object of the present paper is to discuss the 
dissection of abnormal frequency-curves into normal curves. The equations for the 
dissection of a frequency-curve into n normal curves can be written down in the same 
manner as for the special case of = 2 treated in this paper ; they require us only to 
calculate higher moments. But the analytical difficulties, even for the case oi n ■=■ 2, 
are so considerable, that it may be questioned whether the general theory could ever 
be applied in practice to any numerical case. 
There are reasons, indeed, why the resolution into two is of special importance. A 
family probably breaks up first into two sjiecies, rather than three or more, owing to 
the pressure at a given time of some particular form of natural selection ; in attempt¬ 
ing to procure an absolutely homogeneous material, we are less likely to have got a 
mixture of three or more heterogeneous groups than of two only. Lastly, even 
where the heterogeneity may be threefold or more, the dissection into two is likely 
to give us, at any rate, an approximation to the two chief groups. In the case of 
homogeneous material, with an abnormal frequency-curve, dissection into two normal 
curves will generally give us the amount and direction of the chief abnormalitjn So 
much, then, may be said of the value of the special case dealt with here. 
A distiuction must be made between the two cases which may theoretically occur. 
If we have a real mixture of two normal groups represented by our abnormal frequency- 
curve, then, theoretically, it is joossible to find the two components, and these two 
component.s must be unique, If they were not unique, a relation of the following kind 
must hold for every value of x ;— 
0-1 v/ (27r) 
(X - 6ir 
2a,^ _j_ 
{'Itt) 
.ArM 
o-3\/ (27r) 
(27r) 
