264 On the Ffect of a Differential Fertility on Degeneracy 
^ „ ,1 7 Hi ho — HJhi 1 , 
Or, finally : k = ^__^ih^_ 
o-x {Hi'-H^'y- 
Tlius cr„ and k are determined, and f being known, we can find yo- In fact 
Thus the constants of the fertility distribution are fully determined. 
(7) Determination of the constants of the fertility distribution, from a know- 
ledge of the fertility of certain grades of the population. For example, when 
cc= —mio-x, let the fertility be y^, and when x = m2<rx, let the fertility be 3/2- 
Then 
2/1 = 2/06 2 ao ) , 
_ 1 /J»2<r^-^'Y 
2/2 = 2/06 2V "'^o /. 
Let a^j cTo = X, k/at) = /a ; we have : 
'2 (log 2/0- 
log 2/1) 
loge 
'2 (log 2/0- 
log 2/2) 
log. 
2 (log 2/0 - log 2/1) , ^2 (log 2/0 -log 2/2) 
log e "'"V log 
LuJ^-^^^ - / 2(log2/o-lo7y7) | 
h + »h (. " V log e V log e j 
(xviii). 
The signs of the roots are arbitrary and must be selected so as to accord with the 
needs of the problem. For example, if = m^ — m, either \m or //, may be 
equal to 
/ 2 (log 2/0 -log 2/1) , / Mlog2/o-log2/2) 
V log e ~ y log e 
and which we choose depends on whether the maximum fertility lies between the 
two grades or outside and beyond the grade of larger fertility. 
The above equations assume that 1/0 is known, it is the maximum average 
fertility of any grade in the community. A third equation to determine 2/0 from 
the average fertility of the whole community is provided on p. 262, Equ. (xii). 
But the equations then become troublesome and can only be solved by approxi- 
mation. It will be found also that small changes in the fertility curve are not 
very influential in modifying the main results to be drawn from the equations 
set forth in this paper. 
