362 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
Hence ,/{8,” — 48,83} will be real or imaginary, according as 7/7 lies outside or 
between the limits 
1 1) 
tt J [eta )at a} 
If +/n lies outside these limits, then the integral of the right-hand side of 
equation (e) is purely logarithmic; if it lies between these limits, the integral is in 
part trigonometrical. 
Since r must be less than n, it follows that the integral must be trigonometrical if 
these limits are respectively = <0 and = >1, «.<., if 
(p + 1/n) (q + Un) = or > F, 
; 1 
or p must lie between } + A) (1 ote = : 
For example, if 1 = 100, then, if p lies between °6005 and °3995, the integral must 
be trigonometrical. If p lies outside these limits, say = ‘7 for example, then the 
integral will be logarithmic if 7/n does not lie between ‘04 and ‘96, z.e., if we draw 
a small or large proportion of the total contents. 
Let us treat the trigonometrical and logarithmic cases separately. 
(12) ‘Case I 8,7 < 487 8:. 
The curve having the same geometrical slope relation is 
log y = constant — aif log (B, + B,« + Bx”) 
ee By 2 tan 72 283% + Bs 
2B; »/ {48:83 — 82°} V/ {4B Bs — Bo} 

Write « for « + B,/28,, changing the origin ; further put a for ,/ {48,8, — 8,°}/(2Bs), 
f for (283), and v for oe. 
integration, 

then we have, y, being a constant of 
Yo ~y tan —) (v/a) 
Uae (1 + a?/a?) 
This frequency curve is asymmetrical and has an unlimited range on either side of 
the origin. It corresponds accordingly to the curve required as Type IV. 
Here 
a = te,/{4(1 + pn) (1 + qn) — (n — 27°}, 
eh n(n — 27)(p— q) 
ia V/{4(1 + pn) (p + gn) — (n — 27)?} 
m=4(n-+ 2). 

