TO THE THEORY OF EVOLUTION. 
11 
— . Ai(N -0 _ _ , /n 2 (N-n 2 ) 
- V N V N 
. . . (xxvii.). 
To obtain r cd we have, if S 77 denotes an error in any quantity rj, 
Sc + S d = Sn 2 , 
cr,r -f cr/ -f 2 a c a d r cd = a, 2 . . . 
. (xxviii.), 
by squaring, summing for all possible variations in e and d, and dividing by the total 
number of variations. 
Hence, substituting 
the values of the standard deviations as 
found above, we 
deduce 
cr c o~a r cd = — ccZ/N .... 
In a similar manner 
SnjScZ = hbM + (Sd) 2 , 
&d&nd dn x — C7 fr(J d Vf, d ~b 
cr d o- n r d , h — d (a + c)/N . . . . 
(xxx.). 
and 
crdO-„r d , h = d (a + 6 )/N . . . . 
. (xxxi.), 
Now 
N r°° 
n \ = /H— e & dx , 
V 27rJ ft 
S>q = - e-*"hh = - NHS h. 
V 27r 
Thus 
v >h = NHo-/,. 
. (xxxii.), 
and similarly 
cr„ 2 = NK a, c . 
. (xxxiii.). 
Hence the probable 
error of li 
•67449 /(J + d) {a + c) 
Hv/N V N 2 • • • • 
(xxxiv.), 
and of k 
•67449 /(c + cl) (a + b) 
Kv/N V N 2 • • ■ • 
(xxxv.). 
They can be found at once, therefore, when H and K have been found from an 
ordinate table of the exponential curve, and a, b, c, d are given. We have thus the 
probable error of the means as found from any double grouping of observations. 
Next, noting that 
Sn x Sn 2 = N 2 HK BhSk, 
we have cr, h cr n r, h , h = N 2 HK c T h <rhr u , 
or 
^ «j « 2 — 'Y'lib 
c 2 
