28 i d'alembert. - 



519. D'Alembert thinks that Daniel Bernoulli might have 

 solved the problem more simply and not less accurately. For 

 Daniel Bernoulli made two assumptions ; see Art. 401. D'Alembert 

 says that only one is required ; namely to assume some function 

 of y for u in Art. 483. Accordingly D'Alembert suggests arbi- 

 trarily some functions, which have apparently far less to recom- 

 mend them as corresponding to facts, than the assumptions of 

 Daniel Bernoulli. 



520. D'Alembert solves what he calls un prohleme assez cu- 



rieux ; see his page 325. He solves it on the assumptions of Daniel 



Bernoulli, and also on his own. We wdll give the former solution. 



Return to Art. 402 and suppose it required to determine out of 



the number s the number of those who will die by the small-pox. 



Let ft) denote the number of those who do not die of small-pox. 



Hence out of this number w during the time dx none will die 



of small-pox, and the number of those who die of other diseases 



/ sdx\ ft) 



will be, on the assumptions of Daniel Bernoulli, [ — d^ — 



Hence, — dco = (— d^ — 



mnj ^ * 



sdx\ ft) 

 mnJ f ' 



,1 r dco dP sdx 



thereiore — = —34. - — . . 



Substitute the value of s in terms of x and | from Art. 402, 

 and integrate. Thus we obtain 



X 



ft) Ce" 



^ e" (/>^ - 1) + 1 



where C is an arbitrary constant. The constant may be deter- 

 mined by taking a result which has been deduced from observa- 

 tion, namely that ^ = 97 when a? = 0. 



521. D'Alembert proposes on his pages 326 — 328 the method 

 which according to his view should be used to find the value of 

 s at the time x, instead of the method of Daniel Bernoulli which 



