K. Pearson 
195 
We see then that the mean and second and third moments must be found to 
determine this curve. From the second and third moments we have ySj, whence by 
equation (xvii) X is determined. The first equation of (xvi) then gives 
Then equation (xv) gives the distance of the start of the curve from the known 
mean. Further since \ = e'^'', we determine c and finally ?/„ is determined by- 
equation (xiii). There is no obscurity or difficulty with the fitting, if we use 
the method of moments. The cubic equation (xvii) is solvable at once eithei' by 
Lill's, Mehmke's or Reuschle's mechanisms. 
But what are the objections ? 
(i) The curve touches the axis at the end of the range. Skew curves 
extremely often cut it at a finite angle. 
(ii) The skewness has a definite direction, which to be logically consistent 
we ought not to neglect, i.e. since X is always >l, d remains always of one sign. 
(iii) Since X>1, rj, the kurtosis, is always positive and the curve can only 
represent platykurtic distributions. It can never give a curve which deviates from 
the Gaussian curve in the direction of the Laplace-Poisson skew binomial for 
p > '2113 < "7887, because this is essentially leptokurtic. 
(iv) The range of skewness given by is very limited. Differentiating we 
find it is a maximum for A, = 1"7200 and this gives = '2075. The Galton-McAlister 
curve cannot therefore describe any curve whose skewness does not lie between 
0 and 2. A cursory examination of the observational results reached, shows that 
the skewness in all kinds of data over and over again exceeds '2. 
(v) /3i and rj are both functions of X only*. Hence there is a relation between 
them or between i] and %. That is to say the kurtosis is determined by the 
skewness. The kurtosis must vanish with the skewness. But experience shows 
that many distributions are sensibly symmetrical and yet have far from zero 
kurtosis, e.g. nasal breadth in English women, etc. etc. 
Finally consider the ratio fii/rj. If we approach the normal curve as the limit 
to a point binomial (p + q)^" we have seen that 
A/^ = (1 - -!■?>?)/( 1 - Qi^q) (xxi), 
and this equals nothing if we take the symmetrical binomial. Otherwise it has 
a finite value depending upon the particular binomial along which we reach the 
Gaussian curve. The Galton-McAli.ster curve, if we make oa/x infinite, but 
aVxVx, — 1 finite, approaches the Gaussian curve. 
* Actually it is 
j3i'» - 12^1^' + ISfift^ + 04ft - 7)-' + 12V- - 307? + 18(3{'ri - Oftij^ - llTftT? = 0. 
1 have to thank my assistant, Mr J. Blakeman, for much aid in the analysis of this section and 
the followinR section of this memoir. 
