TO THE THEORY OF EVOLUTION. 
5 
For this obviously gives (x.). Assume it true for v H _ Y and v n _ 2 , then 
xv n _ x — {n— l)v „_ 2 = x n — 
(n - 1 )(n ~ 2) (n - 1 )(n- 2) (n - 3) (n - 4) ^ ; _ 4 , 
2 I 1 
'x n ~ + 
92 I 9 
a:" 
/ i \ „_o , ( n — 1) (n — 2) (n — 3) , 
— (n — l)x n 3 -f- -mu- ~x 11 
2 11 
. < n ~ 1 ) -y’i —2 , n(n-l)(n- 2) Q - 8) 
- iXy v) I 1 lb r)2 | IT) tb 
= 
Thus the expression (xv.) is shown to hold by induction, the general terms bein 
(- 1 )’ 
(n — 1) (n — 2) . . . (n — 2r + 1) fn — 2r 
2 r ~ l \r — 1 
2r 
+ 1 ) x n 
_ (- ~ ^ ~ ~ 2r + X) x »-*r 
£' V 
or the general term in u n . 
We notice at once that 
Thus, by (xii.) 
dVn 
clx 
= nv n _, 
(xvl). 
dV n _-y 
v n = -t~. 
dx 
or 
Multiply by e *** and integrate 
| e~ ix *v„ dx ~ | xe~* x 2 v H _ l dx — | e - ^ 2 dx. 
Integrating the latter integral by parts, we have 
j v n e~ lz2 dx — — e~ ix ~v n _ 1 , 
V ” = Tfrr \ t V - e ' Vdx = 
Now —7= e * /t2 can be found from any table of the ordinates of the normal curve, 
v 27 r 
e.g., Mr. Sheppard’s, ‘ Phil. Trans.,’ A, vol. 192 , p. 153 , Table I. # We shall accord¬ 
ingly put 
.(xvii.), 
H = — "-i* 
s/ 2 
e °~ 
K - - - O 2^~ 
±V ~ V / 2 7 r e 
and look upon IT and K as known quantities.' 
* For our present purposes the differences of Mr. Sheppard’s tables are occasionally too large, but the 
following series give very close results :— 
Let 
X, = 
= \/\ - + - + - = jy** 1 i,: v( iv ->, 
\ = by (v.). 
