DANIEL BERNOULLI. 225 



versally admitted. Since the introduction of vaccination, the 

 memoir of Bernoulli will have no practical value ; but the mathe- 

 matical theory which he based on his hypotheses is of sufficient 

 interest to be reproduced here. 



402. Let X denote the age expressed in years ; let f denote 

 the number who survive at that age out of a given number 

 who were born ; let s denote the number of these survivors who 

 have not had the small-pox. Assume that in a year the small- 

 pox attacks 1 out of every n who have not had the disease, 

 and that 1 out of every m who are attacked dies. 



The number of survivors who have not had the small-pox 

 continually diminishes ; partly because the small-pox continually 

 attacks some whom it had previously left unattacked, and partly 

 because some persons die of other diseases without ever being 

 attacked by the small-pox. 



The number of those attacked by the small-pox during the 



sdx 

 element dx of time is by hypothesis — - : because we suppose 



o sdx 



- to be attacked in one year, and therefore in the element 



n n 



dx of a year. The number of those who die of the small-pox is 



sdx 

 by hypothesis ; and therefore the number of those who die 



^dx 

 of other diseases is — d^— - — . But this last number must be 



mn 



diminished in the ratio of s to f, because we only want the 



diminution of those who have not yet had the small-pox, of whom 



the number is s. 



Thus „ds = —-i(d^-^--). 



n g V 7)inJ 



This equation is to be integrated. We have 



s^dx 



• 



, _ O..C ..^^ ^..^ dx 



therefore 



mn 



15 



