PROF. KARL PEARSON ON SKEW VARIATION. 
433 
which must therefore be either zero or a positive quantity. Thus we see that the 
whole area covered by my frequency curves is limited above by the line ft—ft— 1 = 0, 
and below by the line 8ft — 15/3 X — 36 = 0. The first line limits all frequency; the 
second line limits my types.* 
(2) Before proceeding further, let us examine the limit to all frequency. Consider 
the line ft—ft —1 = 0. 
The form of the curve isf 
Now, 
where 
therefore 
Hence 
V»i 
m' 2 —rn' + e — 0, 
r = 6 (ft-ft-1) 
3ft-2ft + 6 ’ 
r — 0 and e = ^r 2 /(l—K 2 ) also = 0. 
m\ + m' 2 — 0 and m\m' 2 =0, or m 1 = — 1, m 2 — — 1. 
The form of the curve is accordingly 
V = 
Vo 
! + -)( 1 
Cli 
oc 
a 2 
or, apparently, U-shaped. Now 
b = b {ft (r + 2) 2 +16 (r+l)} 1/2 
= <r {ft+ 4 
and is finite. But 
?/o 
r (m l + 'm 2 + 2) 
b (wii+ m 2 ) mi+m2 B (m 1 +1) B (m 2 +1) ’ 
_ N (m 1 +1) ( m 3 + l) m 1 mi m 2 CT2 (m l + 2) (m 2 + 2) 
T (m 1 +m 2 + 4) 
But 
b m 1 + m 2 + 2 (m 1 + rn 2 ) mi+m2 (m 1 + m 2 + 3) B (mj + 3) x F (m 2 +3) 
_ N li it , (wfi+^^+l) 4 x r (2) 
b ilDm 01 mi+ m 2 +2 F(2)xF(2) 
limit of K + l) K+l) = 
wii -\- tyi 2 T 2 Ax A 2 
* It is not accurately correct to say it limits my types of skew curves. What it actually does is to cut 
off an area in which the probable errors of the constants of Types IV. and VI. curves can be very great. 
The curves may give a good fit, but the constants cannot be cited as characteristics of the frequency 
distribution as they are unstable. 
t The notation throughout is that of my original ‘ Phil. Trans. 1 memoirs of 1895 and 1901. 
