156 THIRD REPORT — 1833. 



tain resistances, which experiment could alone determine. The 

 accuracy of the general conclusion was affected by several 

 assumptions, namely, the perfect fluidity and sensibility of the 

 mass, which was neither affected by friction nor cohesion, and 

 an infinitely small thickness in the edge of the aperture. 



Daniel Bernoulli, in his great work, Hydrody namica , seu de 

 Viribns et Motibus Fluidorum Commentaria, published at Stras- 

 burgh in the year 1738, in considering the efflux of water from 

 an orifice in the bottom of a vessel, conceives the fluid to be 

 divided into an infinite number of horizontal strata, on the fol- 

 lowing suppositions, namely, that the upper surface of the fluid 

 always preserves its horizontality ; that the fluid forms a con- 

 tinuous mass ; that the velocities vary by insensible gradations, 

 like those of heavy bodies ; and that every point of the same 

 stratum descends vertically with the same velocity, which is 

 inversely proportional to the area of the base of the stratum ; 

 that all sections thus retaining their parallelism are contiguous, 

 and change their velocities imperceptibly ; and that there is 

 always an equality between the vertical descent and ascent, or 

 vis viva : hence he arrives, by a very simple and elegant pro- 

 cess, to the equations of the problem, and applies its general 

 formulae to several cases of practical utility. When the figure 

 of the vessel is not subject to the law of continuity, or when 

 sudden and finite changes take place in the velocities of the 

 sections, there is a loss of vis viva, and the equations require 

 to be modified. John Bernoulli and Maclaurin arrived at the 

 same conclusions by different steps, somewhat analogous to the 

 cataract of Newton. The investigations of D'Alembert had 

 been directed principally to the dynamics of solid bodies, until 

 it occurred to him to apply them to fluids ; but in following the 

 steps of Bernoulli he discovered a formula applicable to the 

 motions of fluid, and reducible to the ordinary laws of hydro- 

 statics. The application of his theory to elastic and non-elastic 

 bodies, and the determination of the motions of fluids in flexible 

 pipes, together with his investigations relative to the resistance 

 of pipes, place him high in the ranks of those who have contri- 

 buted to the perfection of the science. 



The celebrated Euler, to whom every branch of science owes 

 such deep obligations, seems to have paid particular attention 

 to the subject of hydrodynamics ; and in attempting to reduce 

 the whole of it to uniform and general formulae, he exhibited a 

 beautiful example of the application of analytical investigation 

 to the solution of a great variety of problems for which he was 

 so famous. The Memoirs of the Academy of Berlin, from the 

 year 1768 to 1771, contain numerous papers relative to fluids 



