TREMBLE Y. 423 



about 86927 drawings there is an even chance that all the tickets 

 except one will have been drawn ; and he j^i'oceeds nearly to the 

 end of the calculation for the case in which all the tickets except 

 two are required to be drawn. 



782. The next memoir is entitled Becherches sur la mortalite 

 cle la petite verole. 



This memoir is published in the Memoir es de VAcad....Be7'lin 

 for 1796 ; the date of publication is 1799 : the memoir occupies 

 pages 17 — 38 of the mathematical portion of the volume. 



783. This memoir is closely connected with one by Daniel 

 Bernoulli ; see Art. 398. Its object may be described as twofold; 

 first, it solves the problem on the hypotheses of Daniel Bernoulli 

 by common Algebra without the Integral Calculus ; secondly, it 

 examines how far those hypotheses are verified by facts. The 

 memoir is interesting and must have been valuable in a practical 

 point of view at the date of publication. 



784. Let m and n have the same signification as in Daniel 

 Bernoulli's memoir ; see Art. 402 : that is, suppose that every year 

 small-pox attacks 1 in n of those who have not had the disease, 

 and that 1 in m of those who are attacked dies. 



Let a^ denote a given number of births, and suppose that 

 a^, a^, a^, ... denote the number of those who are alive at the end 

 of 1, 2, 3, ... years : then Trembley shews that the number of per- 

 sons alive at the beginning of the x^^ year who have not had the 

 small-pox is 



i-i+i(i-ir 



m 711 V 'nJ 



For let h^ denote the number alive at the beginning of the a^"' 

 year who have not had the small-pox, and ^^^^ the number at the 

 beginning of the {x + 1)*^ year. Then in the x^^' year small-pox 



attacks — persons ; thus h^ (l j would be alive at the begin- 

 ning of the next year without having had the small-pox if none of 

 them died by other diseases. We must therefore find how many of 



