438 
PROF. KARL PEARSON ON SKEW VARIATION. 
This leads on integration to 
V = Vo 
( 17 (v 3+/3, + y /3;) + x \ V 
■mt (-\/3 + fa — \ fa)—x 
I term this Type XII., or the R-line J-curve. The origin is the mean, the range 
from x — —or (vS +/3 1 +\ fa) to <r (\ 3 -t- /S x — v fa). It separates J-curves—so long as 
we are above the line 2 /T—3/3, — 6 = 0 —for which /• — 2 is positive from those in which 
r—2 is negative. But r—2 — + Hence below the R-line the positive oq is 
greater than the negative nr,, but above this line the positive m y is less than the 
negative m 2 , i.e., the upright of the J is emphasised at the expense of the horizontal 
part, while below the Rdine this condition is reversed until on the biquadratic the 
infinite ordinate of the J-upright is replaced by a finite ordinate. 
I propose now to consider a little in detail the nature of these new types of frequency 
and the manner of fitting them to actual data. 1 have dealt above sufficiently fully 
with “block-frequency and its criterion faj—fa — l =0 and therefore need only 
consider Types VIII. to XII. 
(4) Frequency Curve, Type VIIL — 
y - y 0 (l+x/a)- m . 
Range, from x — 0 to x — — a.* 
IP is clearly the value of the ordinate at x — 0, re., the finite ordinate at the tail. 
We easily deduce if N be the total frequency y 0 = N (l — m)/a, and taking the 
origin at n — — a, 
x — y.\ = ct (1 — m)/(2 — rn), fa, = a 2 (1 — m )/(3 — m), 
fa., = a s ( I — m)f ( 4 — m), fa x = a 4 (1 — m)f( 5 — m ). 
Hence for the moment-coefficients about the mean 
a-~ = /JL-, = o' (I — m)/{ 3 — in ) (2 — in )“ [■, 
/.<•: — 2ct"m (l — (2 — inffa 
U-i — 3« 4 (l — m) (4 — 5»H -3?n 2 )/{(o — m) — — (2 — in) 
These lead to 
n 
fa 
4 m 2 (o — m ) 
(l — in) (4 — mf 
fa 
_ 3 (3 — m) (4 — bin + on fa 
(l — o?) (4 — m) (5 —in) 
< dearly rn could be found from the value of fa by solving the cubic equation 
ntf (4 — fa) + nr (9/3, — 1 2) — 2ifam+ 16/3, = 0, 
* Of course, whether a is really positive or negative will depend on the sign given to x, or the direction 
of the ''-axis. 
