Problem in Age-Distribution. 437 



4. From the nature of the problem p(a) and /3(a) are never 

 negative. It follows that (4) has one and only one real 



root r, which is ^ 1, according as 



^ 7 {3(a)p(a)da=l (5) 



Jo <• 



Any other root must have its real part less than r. For 

 if ?\ (cos 0-fi sin 0) is a root of (4), 



1= c?mm cosa da (6) 



Jo r i 



It follows that for large values of t the term with the real 

 root r outweighs all other terms in (3) and B(i) approaches 

 the value 



B(*) = Ar* • . (?) 



The ultimate age-distribution is therefore given by 



F(a,*)=Ap(fl)r^ a (8) 



= &p(a)e r '<t- a \ .... (9) 



Formula (9) expresses the "absolute" frequency of the 

 several ages. To find the " relative " frequency c (a, t) we 

 must divide by the total number of male individuals. 



c(a f)== F(a,Q _ Ap(a)e r ' {t - a) _ p (a)e-"* 



f F(a, 0*» A/« I e - r ' a p ( a )da f e' r 'y(a)da 

 Jo Jo Jo 



= be- r ' a p{a), (10)* 



where 



i=|Vv(«)^ W 



The expression (10) no longer contains t, showing that 

 the ultimate distribution is of " fixed'"' form. But it is also 

 "stable;" for if we suppose any small displacement from 

 this " fixed " distribution brought about in any way, say by 

 temporary disturbance of the othemcise constant conditions, 

 then we can regard the new distribution as an "initial" 

 distribution to which the above development applies : that is 

 to say, the population will ultimately return to the "fixed" 

 age-distribution. 



* Compare Am. Journ. Science, xxiv. 1907, p. 201. 



