MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 369 
This gives 7, then : 
Ted 


52 — Hs (Gar) 7 eni@ae AE ae GG EP ab) 
— = ea) ? 
€ 4 
or 
i= Va {Bi (7 + 2)? + 16 (r +1} | 
Since 
r=, +>, €=M 1M, 
m’', and m’, are roots of 
m? —rm +e=0. 
Thus m, = m’, — 1 and m, = m’, — 1 are determined. 
Further, a, + a, = b, a/dg = m,/m,, and v = m,/a, are all determined. 
Lastly : 
Yo = NM." M.”/(m, + My)™*™, 
and 
a = bn T(m, +1) 0 (m, + 1)/P (m, + m, + 2), 
give: 
eee My" Mg"? T (am, + my + 2) 
Y= F (m, + m,)m*™ T (m, + 1) (m, +1)’ 


which completes the solution,* if a Table of I functions is to hand. 
Remarks.—It is clear that the solution is wnzque. 
It is necessary in order that the solution may be real, that m’, and m’, should be 
real or 7?>4e. Hence, if « be negative, there is certainly a solution, because 1 is 
always real. The solution forms, however, one of the sub-types referred to in our 
Art. 13, (ii) and (iii). 
If « be positive, we must have 7*/e — 4 positive, or 

Bi (3 + By? 
(+ 38; — 28,) 48, — 38) 
Now it is easy to prove that for any curve 48, — 38, or 44. — 3p;° is positive, 
for py is always greater than p,°. 
Thus, we must have 
6 + 3B, ri 2B, > 0, 
or 
2 pea (Bprg” — py) + 3pyg° > 0. 
* Very often with sufficient accuracy we may take: 
io & (mt ttg +1) A (0m + 19) 5 Gant ~ mm ~ md 
aan / (2m my Mg) 
MDCCCXCV.—A. a) 1B 

