PROF. KARL PEARSON ON SKEW VARIATION. 
449 
characters. Below the point f3 2 = 1, descending the (3 2 - axis, the two concentrated 
frequencies expand into a symmetrical U-curve. This is Type Ily with the equation 
y = y 0 (l —x 2 /a 2 )~ m 
and the criterion /3 1 = 0, (3 2 < 1'8. 
Here* 
m = £(9-5&)/(3-&), 
a 2 = a 2 .2f3 2 /{3-(3 2 ), 
and 
? N F (f-m) 
\/2ir(j T (l — m) x/f -—m 
When (3 2 = 1*8, m — 0, and we reach the “rectangle-point ” R. Here y 0 = N/(2 a) 
and a = a I v / 3 . 
Samples of three individuals from a population whose individuals carry two 
uncorrelated characters give a symmetrical U-frequency for the coefficients of 
correlation of those characters in triplets of individuals. In this illustration (3 2 = 1'5. 
Samples of four individuals from the same population give a rectangle for the 
frequency distribution of the coefficients of correlation. Passing still lower down the 
axis of symmetrical frequency the type is now Type II L , or the limited range 
frequency curve 
y = y, (l-x 2 /a 2 ) m 
and the criterion is /3 X = 0, /3 2 > 1'8 < 3. 
In this range m increases from 0 to o°, and 
m = i(5/3 3 -9)/(3-&) 
a; = P. 2,a,/(3-,ffi), 
N r(# + m) 
= - — - - > . . . 
\/27ro- r (l +n;-) \/f + m 
We see that the range grows greater as m approaches infinity, or /3 2 = 3, when we 
reach G the Gaussian point (/3 X = 0, /3 2 = 3). 
If samples of n individuals he taken from an indefinitely large population in which 
the individuals carry two uncorrelated characters, then if n be 5 or over, all the 
frequency curves of the correlation coefficients of these samples are of Type II L , only 
approaching the Gaussian when n is very considerable indeed. For example when 
n=25, f3 2 = 27692, and the frequency is still a good way from the Gaussian. 
When n = 400, (3 2 = 2'9850, it is thus fairly close to it, but is not coincident. 
* It is, perhaps, worth noticing that for /3. 2 = 15/7 we obtain the ordinary parabola as a special type of 
frequency-curve. 
