We may suppose that this risk becomes effective in the 

 constant proportion q only of the whole, q ordinarily being 

 less than unity, and in no case greater than unity. So long 

 as the first birth-rate and infantile mortality remain con- 

 stant, the number X at risk, assuming, as we may without 

 sensible error the absence of multiple births, is 



(3) N = P-B + qM. 



When a change of rate of infantile mortality supervenes, 

 the number of births must as stated be obviously affected 

 if the reproductivity remain constant, for the number at 

 risk X' under the new conditions is 



(3a) N' = P - B :'+ qM'. 



That the reproductivity of these two groups X and X' shall 

 he identical, it is necessary that the ratio of B to the first 

 shall be identical with that of B to the second, that is B/X 

 — B X . Hence taking the reciprocals of these quantities, 



P + q M , _ P r q M _ . 



' /{ B' 



But H/P and B'/P are the initial birth-rate and the birth- 

 rate as changed by change of infantile mortality, say ft and 

 ft', hence after throwing out the unit on each side this last 

 equation becomes 



(5) i + q*= t+«X 



(5a) f= 1 + /?" (/'-/*') 



This is the fundamental relationship between the rates 

 considered. This formula may be simplified in practical 

 applications. Since the quantity to be algebraically added 

 to unity is in all cases very small, it will be sufficiently 



(6) ft' = ft { 1 + q/i(/*' - vi- 

 ndicates that the change is sensibly a linear < 



