450 
PROF. KARL PEARSON ON SKEW VARIATION. 
After we have passed the Gaussian point we obtain curves of unlimited range of 
Type VII., of which the equation is 
V = ?/o( l+x 2 /a 2 )~ m , 
The range of ft, is from 3 to oo and 
m — |-(5/3.,—9)/(/3, —3), 
falls from infinity to 2‘5 ; while 
a 2 = a- 2 . 2&/(&-3), 
t = _N_ P(m) 
\/27 to- F (m—■§-) \/{m— f)‘ 
Illustration of curves of Type VII * are not infrequent in biological statistics. We 
see that the Gaussian is a mere point in an infinite range of symmetrical frequency 
curves, and a single point in a doubly infinite series of general frequency distribu¬ 
tions. 
Now let us consider the asymmetrical frequency curves displayed on the Diagram. 
If we approach from the “ impossible area ” we reach on the B-line the first available 
type of frequency—the alternative concentrated blocks. At one end of the B-line 
we have two equal isolated frequencies, and at the other a single isolated frequency. 
Crossing the B-line we reach the area of limited range U-shaped curves, i.e., Type 
I U5 which has for its equation : 
jj = y 0 (1 +x I aft)-" 1 ' (l —x/a 2 )~ m \ 
This U-area extends as far as the upper branch of the loop of the biquadratic, the 
asymptote of which, 24/3 2 —27/3j — 38 = 0, is indicated by a broken line. In U-shaped 
frequency curves both m x and m 2 are necessarily less than unity, for their product 
is e—r+1, which is less than unity and positive above the upper branch of the 
biquadratic (i.e., e— r+1 = 0). Type I L - is fitted as Type I. (see ‘ Phil. Trans.,’ A, 
vol. 186, p. 367), and has been illustrated by me (‘Boy. Soc. Proc.,’ vol. 62, p. 287), 
by fitting curves of frequency to cloudiness. The frequency curves for the correlation 
coefficients of samples of three drawn from a population whose individuals have 
two characters of any degree of correlation are also skew U-shaped frequency 
curves, although their algebraic form has not the above simplicity. 
* Type II L was discussed in my first memoir, ‘Phil. Trans.,’ vol. 186, p. 372. Type II. T and Type VII. 
are briefly referred to in ‘ Biometrika,’ vol. IV., p. 174, but, unfortunately, with some rather disturbing 
misprints. They are correctly placed on Rhind’s diagram, 1 Biometrika,’ vol. VII., p. 131, but the 
formulae for fitting are not given. The formulae have been given for many years in lecture-notes, and 
the curves have been frequently used. 
