MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 363 
Special cases. (i.) Suppose r/n = x, and v very large, then 

9 
m/a? = = = — = a), Say, 
DCSE aay) 
Lo Gi 
Sip Se ee 
¢ (pa — (4 -x)”) 
Thus we have 
y —_ Yoe = a,x? — oar 
which reduces to the normal type by a change of origin. It is important to notice, 
however, that the standard deviation of this normal type 
= /(1/2m)) = e/ {nr (pa — (4 — x)’)}; 
and is very different from the value ¢,/{(r + 1) pq} = 4ce,/ (npq X 4x), nearly, which 
is the usual form. Only when we put p = q = 3 and make y small do they agree. 
We thus conclude: That the normal form may fit a chance distribution, but it does 
not follow that the standard deviation is of the binomial type generally assumed. 
(i.) Suppose y = 3, corresponding to the withdrawal of one-half of the contents of 
a vessel, then 
Y = Yo (1 + 2*/ay")~, 
ly = 3C/{(1 + pn) (1 + gn)}. 
This is an unlimited and symmetrical frequency curve approaching more and more 
where 
nearly to the normal form as we increase 7. It has, however, a standard deviation 
= 4¢,/(npq), while the normal curve would give 3c,/(npq X 2). 
(iil.) Suppose p = q = 3, we again reach the form 
y=y (1+ e/a"), 
eA sal) 
Make vn infinite and we have again the normal type, but a standard deviation of 
the form $¢,/ {nx (1 — x)}, only approaching the usual value when y is small. 
We postpone until we have discussed the remaining types the problem of fitting a 
curve of Type IV. to a series of observations. 
(132), Case Ll 8.7 > 46) By, 
Let a and a, be the roots of B, + B,« + B,x?= 0. Then the curve having the 
same geometrical relation for its slope is 
where 
d (log Yy) ab L 
dg ~~ B,(«% —a,) (%@ — a) 
= lead 
ae Bs (4 — 4) dix 

{a log (w — a) — a, log (w — ay)$, 
3A 2 
