23 i DANIEL BERNOULLI. 



Suppose at the end of the time t the quantities of the white 

 fluid in the three vessels to be u, v, w respectively. We obtain the 

 following equations, 



du = kdt (w — y), 



dv = kdt (u — v), 

 dw — kdt {v — w)y 

 where Zj is a constant. 



Daniel Bernoulli integrates these equations, by an unsym- 

 metrical and difficult process. They may be easily integrated by 



the modern method of separating the symbols. Put i) for -7- ; thus 



at 



{D + ^) w = kw, (J) ■\-k) v= kuj (I)-\-k)w = kv, 



therefore (D -\-Tcf u = Hu. 



Hence u = e"^* [Ad"^ + Be""^' + Ce^*^'}, 



where A, B, G are arbitrary constants, and a, /3, 7 are the three cube 

 roots of unity. The values of v and w can be deduced from that of 

 u. Let us suppose that initially u — h, v = 0, i^ = ; we shall find 



that A =B= C=^, so that 



o 



Laplace has given the result for any number of vessels in the 

 Theorie...des Proh. page 303. 



421. Now it is Daniel Bernoulli's object to shew, that when x 

 and n are supposed indefinitely large in the former problem its 

 results correspond with those of the present problem. Here indeed 

 we do not gain any thing by this fact, because we can solve the 

 former problem ; but if the former problem had been too difficult 

 to solve we might have substituted the latter problem for it. And 

 thus generally Daniel Bernoulli's notion is that we may often ad- 

 vantageously change a problem of the former kind into one of the 

 latter kind. 



If we suppose n and x very large we can obtain by the Bino- 

 mial Theorem, or by the Logarithmic Theorem, 



