TO THE THEORY OF EVOLUTION. 
13 
It remains now to determine y 0 , y 1; and y 2 . 
By Equation (i.) 
d =f{r , h, k) = 27rv/ ^— i\ h j^e-l-Y^^dxdy, 
_df _ 
7l dh 
N i 
„ - -i-17i 2 + ?/2 - 2rhv') .7,, 
■N* __ i/ t 2 f i (y—rfi)* j 
= 2-Vl-^ ),« ^ 
N" r 00 
= - :H Lv‘*' *. 
. . . (xliv.), 
where 
Thus 
n k — rh 
yr-r 3 ’ 
4 NH) =- fe C*"'. * - ^ C*"' *) 
= 'p-i \ . 
. . . (xlv.). 
Similarly 
y 2 /( NK) = f-l. 
. . (xlvi.). 
Here 
+•- ✓ “*• ~ dZ - ■ 
. . . (xlvii.), 
where 
n h — rk n k — rh 
— v/l - r 2 ’ v/l -r 3 * ‘ • 
. . .(xlviii.), 
and thus i/q and xfj. 2 can be found at once from the tables when (3 1 and are found 
from the known values of r, h, k. 
Lastly, we have from Equation (xxi.) 
or 
Thus* 
where 
d 
N 
ad — be 
— e W + W) 
N 2 HK 
(d + b)(d + c) 
N 3 
fll dr, 
Jo 
+ L l 0 
U dr. 
y 0 = df I dr = NU, 
7r 
7o/ N = Xo> 
2ir 1 — i 
f, C 2(1 — )' 2 ) 
(7i 2 + l 2 - 2WU) 
(xlix.) 
a value which can again be found as soon as r, h, 1c are known. y 0 = y 0 N is clearly 
the ordinate of the frequency surface corresponding to x = h, y = k. 
Substituting in Equation (xliii.) we have, after some reductions, 
* By Equations (ii.) and (iii.), cl + b and d + c are independent of r. 
