128 Mr. Gwyther on the 



was not attacked on the score of not representing the actual 

 facts of the case. 



Passing on to Bernoulli's second Investigation — 



Theorem II. — To find the ratio of the number of those 

 who would be alive at any age, If death from small-pox was 

 prevented, other circumstances remaining the same, to the 

 number who would be alive on the hypothesis of the pre- 

 vious theorem. 



The other letters retaining their original significations, 

 let K denote the numbers living at the age x under the 

 changed circumstances. 



The total mortality during the time dx will now be 



-l{dt, 





which we must write ~ d'Q. 



, dii dl __ sex 



substituting for sjt, and Integrating we get 



4 — X v. 



{m- i)e" + I 

 and thus the required ratio has been found. 



Now Bernoulli's object was to show to what extent life 

 would be saved by adopting the system of inoculation, but 

 we may apply his results to the consideration of other cir- 

 cumstances not foreseen by the author. Take the two most 

 extreme cases to make a comparison. Imagine a locality 

 in which small-pox Is epidemic for x years, and that, in 

 another locality having the same number of Inhabitants, 

 from superior sanitary or other reasons small-pox does not 

 appear for x years, but that the inhabitants have no immu- 

 nity against small-pox in case it should be introduced, we 

 see from what I have proved above that the ratio between 

 the numbers subject to small-pox In the two places would be 



