Change of Incidence of Small- Pox. 125 



at that age by g, the number of these survivors who have 



not had small-pox by s : suppose that during the unit of 



time in which x is measured, on the average out of n persons 



one takes small-pox, and that out of m persons who take 



small-pox on the average one dies, and let us suppose that 



m and n are constants at different ages. 



Then the number of those who take small-pox durin^r 



Sox ^ 



the time dx is — , and the number for whom it ends fatally 



^^ '^n Consequently those who die from causes other than 

 small-pox is -^5-^. Taking the proportion of these 

 deaths which correspond to the number s who have not had 

 small-pox, we get 



set, s-dx 

 t, uint 



Then -ds denoting evidently the number who take 

 small-pox and do not die, during the time dx, we have the 

 equation 



_ , _£££ sdt s-dx 

 n t, mnt, 



Although this equation contains three variables, s and 'i 

 only appear in the form of the ratio s'X-=q and we get 



inndq 



=dx 



mq - I 



in 



+ I 



And we can determine the constant by supposing that at 

 birth s = ^,q=\, hence 



m 



(m- i)f^ + I 

 From this relation Bernoulli easily forms theoretical tables 

 which it appears that he thought fairly represented the 

 small-pox mortality at the time. Putting m and n each 

 equal to 8, he considers that his tables give good results. 



