TO THE THEORY OF EVOLUTION. 
3 
Further, 
b -f- d — . 3 - ( e “ <rp ckc, 
V / 27TO- 1 J/ t ' 
N f" 
= _i . . . 
V 27T J a 
and 
N f°° 
c+rf= v /^L e . 
• • • • 
■ . . . (iv.), 
(8 + S) 7 (c + ' ) = a/IC^ 
.(v.). 
Thus, when a, b , c, and cl are known, h and h can be found by the ordinary table of 
the probability integral, say that of Mr. Sheppard (‘ Phil. Trans.,’ A, vol. 192. p. 167, 
Table VI.*). The limits accordingly of the integral for d in (i.) are known. 
Now consider the expression 
- 2 e-*rr^ + 2,2 ' 2rxy) = U, say, 
x/1 - r 
and let us expand it in powers of r. Then, if the expansion be 
(vl), 
U = e-^ + *(«o+ ?£' + ¥+ • • . +n?+. • 
we shall have 
|1 
u n — e i( * 2 + y2) 
n 
i d ’ 1 U 
\ dr n J r = 0 
. . . (vii.), 
. . (viii.). 
Taking logarithmic differentials, we get at once 
r/IJ 
(1 — r 2 ) 2 = {xy + r(l — x 2 — y 2 ) -f- I'^xy — p 3 }U. 
Differentiating n times by Leibnitz’s theorem, and putting r = 0 , we have, after 
some reductions 
Hence we find 
u n+l = n{2n — 1 — x 2 — y % )u n _ x 
— n(n — 1 ) (n — 2) 2 u„_ 3 
+ xy{u H + n{n — 1 )u n _ z ] 
Uq — 1 
u 1 — xy 
u 2 = (X 2 - 1 ){lf - 1) 
u 3 — x(x 2 - 3) y{if - 3) 
u. 4 = (jc* - 6x 2 -f 3) ( 7 / - 6y 2 + 3) 
* See, however, foot-note, p. 5. 
B 2 
(ix.). 
(x.) 
