CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 
447 
Put = 2^, and we find 
fee 
g-.- ,72 
. 0 
^y^yn-P+ir{2)-n-l) . . 
Thus: a = 7 /oyi-^r(p- 1) . 
=y/ {p — 2) 
= rV (p - -) (p - 3) 
H = yV (p — (p - 3) (79 — 4) 
= yV (p - 2) (p - 3) (73 - 4) (p - 5 ) 
Transferriuo' to the centroid we find 
O 
■ (?;-2p(7> - :i) 
_ __ 
{p — -Zf{p — 'd){p--y) 
_ 3(p + 4)7^ _ 
- 3)(p-4)(p-5)i 
= P3VP2^ 
16 (jo - 3) 
(p - 4)2 
A 
/ 2 _ 3 (jj + 4) (y> — 3) 
P4//^2 — _4) 
Eliminating p between and /So we find alter some reductions : 
or, 
/8i (A + 3)2 = 4 (2^, - 3/3, - 6) (4^o - 3/3,) 
ACo = 1. 
(v.). 
(vi ). 
(vii.). 
(viii.). 
(ix.). 
(X.). 
(xi.) 
Clearly, since this is the condition for Type V., that transition curve is none other 
than the curve obtained by making the denominator of the right-hand side of the 
dilferential equation have equal roots. The curve is clearly of considerable interest, 
and its existence had not been noticed in the previous series of frequency curves. 
The manner of fitting it is now easily described. 
Equation (ix.) gives us a quadratic to find p — 4 : 
/ .1 1 Ij 16 / .. ^ 
(p - 4)— — (p - 4) - - = 0.(xii.). 
The positive root of this is the required solution. 
