PROF. KARL PEARSON ON SKEW VARIATION. 
453 
They have been considered on p. 443 of the present memoir. Their full theory 
is precisely that of curves of Type VI. in general, discussed in the first supplement to 
my memoir on skew variation (‘ Phil. Trans., ’ A, vol. 197, p. 448, et seq.). The only 
point to be emphasised is that the q 2 of Equation XIX. of that memoir in this area 
is negative and less than unity. The treatment is identical. 
Below both the Type III. line and the biquadratic, we have a space bounded by the 
cubic 
4(4/32-3/3,) (2/3 2 -3/3,-6) - /3, (/3 2 +3) 2 . 
This is the area of Type VI. proper, i.e., 
y = y 0 (x — a) q2 /x 9 ' 
with range from x = a to x = 00 , q 2 <Cq 1 being positive, and is fully discussed in the 
memoir jnst cited. 
The area of Type VI. is limited by the above cubic along which Type V., or, 
y = y,x- p e~ ylx 
from x — 0 to x — co, describes the frequency. Its full consideration will be found in 
Phil. Trans.,’ A, vol. 197, p. 446, et seq. Below the Type V. cubic we reach the 
area of Type IV. curve, or 
y = y 0 e~ vtan ' l{x/a) /( 1 +(xja 
This lias unlimited range in both directions and its treatment is fully discussed in 
my first memoir (‘ Phil. Trans.,’ A, vol. 186, p. 376, et seq.). Theoretically, Types IV. 
and VI. describe all types lying below the line 2/32 — 3/3, —6 = 0. The objection to 
their use lies in the increasing probable errors of their constants, however good their 
general fit may be. To warn the statisticians of this, the line 8/3 2 —15/3, —36 = 0, is 
drawn on the diagram and the area below it is marked “heterotypic area.” I use 
this term to signify that it is doubtful whether my skew-frequency curves, depending 
only on the first four moments, can adequately describe distributions of types falling 
below this line; they require the use of the fifth and higher moment coefficients. 
Their occurrence in practice, however, must be rare. 
It will be noticed that the line /3 2 —/3, — 3 = 0 is drawn through the Gaussian point. 
This is the relation which must be satisfied in the case of Poisson’s exponential limit 
to the binomial. Hence, in the case of a distribution with /3„ /3 2 , near this line, it is 
worth while investigating whether the “ law of small numbers ” is appropriate. Above 
this line every real binomial distribution, i.e., cases of p and q both positive and less 
than unity, and n positive (taking the binomial as (p + q) n ) must lie, for 
/3 2 — 3 _ I — 6pq } 
/3, 1 — 4 pq' 
3 Q 
VOL. CCXVI.—A. 
