190 
Skeit) Variation, a Rejowder 
Now foi" the Galton-McAlister curve 
(X + 2)= 
\^ + + 6\ + 6 ' 
and this approaches the limit 9/16, its maximum vahie, when X, approaches unity. 
If we take (1 — 4pg)/(l — Qpq) = 9/16, we get imaginary values for p and q. Thus 
while the normal curve itself gives an indeterminate value for /3i/'j? = 0/0, and as 
Laplace has shown describes with fair accuracy any slightly skew binomial with 
large power, the Galton-McAlister curve cannot describe even approximately any 
skew binomial, however near to a normal distribution. 
On all these grounds we see that the " law of the geometric mean " fails to 
supply the fundamental need of describing the modal difference, the kurtosis and 
the skcwness of actual frequency distributions. It cannot describe these physical 
characteristics of the frequency. 
C. (iii) Fechners Double Gaussian Carve*. 
We have noted that the Gaussian curve was first deduced by Laplace to 
represent a finite number of the terms of a binomial expi-ession, and that Gauss 
deduced it on hypotheses which amount to the following: 
(i) The arithmetic mean is the most probable value. 
(ii) Deviations in excess and defect of the mean are equally probable if of the 
same magnitude. 
(iii) The facility of an increment is the same for all values of the character. 
Now every one of these assumptions is negatived when the double Gaussian 
curve is used, and yet the Gaussian curve wJiich is only deduced by aid of them is 
adopted to describe what conflicts with its fundamental axioms. This proceeding 
is the reverse of logical. However, if the double Gaussian curve be adopted, there 
is absolutely no reason why we should adopt the rough process by which Fechner 
determines the mode and obtains the constants of the distribution. The fitting 
by my method of moments is perfectly straightforward, and as it leads to 
the points we have to consider it will be indicated here. Let the two half 
curves be : 
1 x-i 
yi — ^— e 2 <r,2 ^ x>Q 
v27ro-] 
1 X,- 
.(xxii). 
y.. = e ia?, x<Q 
Then, since the modal value is common, aijn-^ = a^^jun. Further, the total frequency 
N =\{ni + n.,). Now write «: = \/2/7r and u = a-i — a., v = aj(To. Then taking 
moments round the mode we easily find : 
" Here again it is historically iucorreet to attribute these curves to Fechner. They had been 
proposed by De Vries in 1894, and termed " half-Galton curves," and Galton was certainly using them 
in 1897. Sec the discussion in Yule's memnir, It. Statist. Soc. Jour. Vol. lx. p. 45 ft xeq. 
