lotka: a natural population norm 293 



For the stable age-distribution this becomes by (1) 



b,n = 6m e-" pm (a) /3,n (>) da (27) 



e-" Pn. (a) I3m (a) da (28) 



an equation which determines r. 



Equation (28) gives rise to two reflections. 



In the first place it can be seen by inspection, that r ^ accord- 



f" > 



ing as I Pm{a) I3m{a) da^=l. This is due to the fact that this last 



integral represents the ratio of the total male births in two suc- 

 cessive generations. 



The second conclusion which we may draw from equation (28) 

 is at first sight somewhat surprising. In that equation we may, 

 without altering its meaning in any way, write the limits of the 

 integral Gi and ^2, instead of and oo , if we denote by ai and a-> 

 the lower and upper limits of the reproductive period. For 

 outside these age limits the function /3(a) has everywhere the value 

 0, so that the terms of the original integral outside these limits 

 contribute nothing to the numerical value of the integral. This 

 being so, the per cent rate of increase of a population in which the 

 stable age-distribution has become established is quite indepen- 

 dent of any factors which may afTect the life of individuals out- 

 side the reproductive age limits — so long as conditions within 

 these limits remain unchanged. Thus, if we were dealing with 

 a herd of cattle, for instance, it is quite immaterial, so far as the 

 effect upon r is concerned, how we slaughter the cattle of the herd, 

 so long as we spare the individuals of breeding age. This is a 

 somewhat surprising result, especially as it applies not only to 

 the superannuated, but also to the young, immature cattle. 



