228 DANIEL BERNOULLI. 



Substitute for s from the result in Art. 402 ; then integrate, 

 and determine the arbitrary constant by the condition that 2=^ 

 when x = 0. Hence we shall obtain 



z me^ 



^ (m - 1) e" + 1 

 Thus as X increases, the right-hand member approaches the 



limit 



m — 1 



406. After discussing the subject of the mortality caused by 

 the small-pox, Daniel Bernoulli proceeds to the subject of In- 

 oculation. He admits that there is some danger in Inoculation, 

 but finds on the whole that it is attended with large advantages. 

 He concluded that it would lengthen the average dur^ation of life 

 by about three years. This was the part of the memoir which 

 at the time of publication would be of the greatest practical 

 importance ; but that importance happily no longer exists. 



407. We shall find hereafter that DAlembert strongly ob- 

 jected to the justness of Daniel Bernoulli's investigations. La- 

 place speaks very highly of Daniel Bernoulli ; Laplace also briefly 

 indicates the method of treating the problem respecting Inocula- 

 tion, but as he does not assume ?/^ and w to be constant, he rather 

 follows DAlembert than Daniel Bernoulli; see Theoiie...des Proh., 

 pages cxxxvii. and 413. 



408. The next memoir by Daniel Bernoulli is entitled De usu 

 algoritlimi infinitesimaUs in arte conjectandl specimen. 



This memoir is contained in the Novi Comm...Petrop. Vol. xil, 

 which is the volume for the years 17C6 and 1767 ; the date 

 of publication of the volume is 1768 ; the memoir occupies 

 pages 87 — 98. 



409. The object of the memoir is twofold. A certain problem 

 in chances is to be solved, which is wanted in the next memoir to 

 which we shall come ; and the introduction of the Differential 

 Calculus into the Theory of Probability is to be illustrated. The 

 reader will see in Art. 402 that Daniel Bernoulli had already really 



