198 
Sleiv Variation, a Rejoinder 
Now : 
^ -45352 -05003 
7/ = /^?, - 3 = — , 
7 T 
and this with the above limitation to tlie vahie of 7 can never become negative. 
Hence the double Gaussian curve is, like the Galton-McAlister curve, invariably 
platykurtic. Now consider the value of 
^ -03662 -11477 -005] 8 
ry ry- ry- 
B 
Hence the ratio — tends as 7 increases to take the value 1-40374. Equating this 
to (1 — 4<2Kj)/(l — Gpq) we sec that the double Gaussian curve approaches the normal 
curve along the particular platykui tic binomial ^) = -80S5, 7 = -1015, or it cannot 
in the neighbourhood of the normal ciu-ve represent any skew binomial but this. 
Lastly it may be shown that /3i has its maximum value when 7 = -36338 or its 
minimum value. Thus we find that the maximum possible value of /Sj is about 
-99. In the same way the maximum skewness is r3230. These values are 
sufficiently high to cover the great bulk of cases, but I have found = 4-071 for 
scarlet fever incidence, = 1-9396 for age of brides who marry men in their 24th year 
and =4-1683 for the distribution of lips in the medusa P. pentata. These 
exceptions suffice to show that the curve is not general enough. 
Summing up we conclude that the double Gaussian curve is not satisfactory 
because theoretically 
(i) It starts by denying the very axioms from which alone we can reach the 
Gaussian curve ; 
and empirically because 
(ii) It can describe no frequency distribution which cats the axis at a finite 
angle, and such distributions constantly occur. 
(iii) It is essentially platykurtic. Therefore it is not available for leptokurtic 
curves, nor even for any but very special skew binomials, i.e. those in which p does 
not lie between -2113 and '7887. As we approach close to the normal curve we 
get nearer and nearer to one definite point binomial, i.e. that in which p = -8985. 
(iv) There is always a relation between the skewness and the kurtosis, or 
these important physical constants are not independent. In particular we cannot 
have any form of symmetry but the mesokurtic. 
(v) The range of /3i and of the skewness is faii-ly large, but frequency 
distributions actually occur markedly outside this range. 
(vi) Lastly, and of much import, the kurtosis can never exceed '8692, or the 
maximum value of ^80 = 3-8692. This degree of kurtosis is exceeded in a great 
number of distributions. Thus in the lips of P. pentata, in tint guessing, in the 
breadth of male English skulls, in the nasal breadth of female English skulls, in 
