176 
Miscellanea 
may be taken as given by/i {x-x) and fo {,>/ — >/). Let NixN-^, be the total number of possible 
matings 
= JI/i (•*■• - x)f.i (y - y) dxdy 
and the total number of offspring N-^ in any array 
I now propose to give the expression for the ni\\ moment coefficient aljout the mean, i.e. jx^", 
of the population of offspring of a given sex. AVe have 
iVi X N. X /V3 X ^„"' = J j J{i 0-3" + ^-^j + « (■*-■ - *-)/3 - j/) x/3 ('«) c^A-rfyc/", 
the integration being extended over the whole of the frequency distributions of father, mother 
and offspring. Thus 
Now X, I) and u being independent we have 
hi — s — t \t 
x/i {x - x)/2 0/ - y)fz {u) dxdydu. 
{X - X)- {x - x) dx = ^'„,_ , _ ( 
Thus 
n - s -t\t cTi" 
.(iv). 
Thus we reach, remembering that (Ui' = jLti" = ^i"' = 0, 
M3 
0-1^ 0-2^ 
;)+M3"' 
, 
16 ■ Vo-]'* o-j'^ a-i' (T.^j 2 
But /ij' = o-i'-^, /m/ = o":;-, and p.{" = 0-3-. Hence we nnist have 
...(V), 
..(vi), 
.(vii). 
.(viii). 
If as usual we take i3i = /ii3^//i-/ and — l^il y-i'' we find from (vi) and (vii), writing s- = /x2" 
^^^ = ('jy{^^r-8(V/3/ + Vft')} (ix), 
^2'^ = w - yq (1^2 + ft" + 6) - 3 '^i 
Whence by the use of (viii) 
.(X). 
/32- = 4|^r- ^ (3,,' + ^/' + 6)| -6. 
..(xi), 
.(xii). 
