372 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
(17.) Generalised probability curve of Type II. Limited Range and Symmetry. 
Y = Yo (1 — x*/a*)™. 
The solution in this case follows very easily from (14) by putting 8, = 0, we have at 


once 
6 (8, — 1) 
Bie Bas aay 
or 
5B,—9 — Sp, — 9p" ; 
oe OG) = FERS) 
Since p, = ar , and clearly e = 7°/4, 
we have b= 24 = 2/ {py (7 + 1)}, 
or 
Gp V/ (248) eae J/2 Hops) ; 
J/(3 = By) J (Bp9" — fy) 
Finally 
ea me T'(2m + 2) 
Jo= b (2m) {TP (m + DP’ 
— & /(3 — B.) T(2m + 2) 
2 VS (2pyBo) 2 {T (am + 1)?’ 
os nN) oe =H TQm+2) 
me 2 popes Dems (m + 1)}? 
Quo, VW 7l' (m+ 1) 
For the normal frequency curve p, = 3p,”, for a symmetrical point-polygon 341.?> py. 
Hence, whenever a symmetrical frequency curve differs from the normal curve on the 
side of the point-binomial, we can better the normal solution by taking a symmetrical 



frequency curve of limited range. 
y=yo(1 _ ae ", 
a 
Since 
and 
m d8,-9 1 
e 4 pty, ply 

if B, = 3, we easily trace the transition from the limited symmetrical curve to the 
normal curve with infinite range. 
Quite apart from the extremely interesting problem of finding the range, it is clear 
that better fits will be obtained for symmetrical distributions by the aid of this limited 
range curve for all cases in which 3p” > py 
