436 Dr. F. R. Sharpe and Mr. A. J. Lotka on a 



2. Let the number of male births per unit time at time t 

 due to the F(a, £)da males whose age lies between a and a + da 

 be F(a, t)/3(a)da. 



If 7 is the age at which male reproduction ends, then 

 evidently 



B(*)= f^Ca, t)P(a)da 



*- o 



= r 7 B(i-a) j p(a)/3(a) £ fa. . . (2) 



Jo 



Now in the quite general case /5(a) will be a function of 

 the age-distribution both of the males and females in the 

 population, and also of the ratio of male Lirths to female 

 births. 



We are, however, primarily concerned with comparatively 

 small displacements from the " fixed " age-distribution, and 

 for such small displacements we may regard /3(a) and the 

 ratio of male births to female births as independent of the 

 age-distribution. 



The integral equation (2) is then of the type dealt with by 

 Hertz (Math. Ann. vol. lxv. p. 86). To solve it we must 

 know the value of ~B(t) from £ = to £ = 7, or, what is the 

 same thing, the number of males at every age between 

 and 7 at time 7. We may leave out of consideration the 

 males above age 7 at time 7, as they will soon die out. We 

 then have by Hertz, loc. cit., 



h=*> al\ \B(a')—\ /3(a 1 )jt)(a 1 )B(a — ajdajctk'da 

 B(0= 2 _i° ^ — . , (3) 



I a/3(a)p(a)xh a da 

 '0 



where « l5 a 2 , . . . are the roots of the equation for a; 



l={ 7 {3(a)p(a)*- a da (4) 



Jo 



The formula (3) gives the value of B(f) for t > 7, and the 

 age-distribution then follows from 



F(a,t)=p(a)F>{t-a) (1) 



