468 PHILOSOPHICAL TRANSACTIONS. [aNNO 1795. 



pies on which the times of emptying vessels are founded, entirely from experi- 

 ment. He was too cautious to trust to theory alone, under all the uncertainties 

 to which he appears to have been sensible it must be subject. He had, in a pre- 

 ceding part of that great work, deduced the general principles of motion, and 

 applied them to the solution of problems which had never before been attempted; 

 but when he came to treat.of fluids, he saw it was necessary to establish his prin- 

 ciples on experiments; principles not indeed mathematically true, like his general 

 principles of motion before delivered, but, under certain limitations, sufficiently 

 accurate for all practical purposes. 



The principle to be established in order to determine the time of emptying a 

 vessel through an orifice at the bottom, is the relation between the velocity of 

 the fluid at the orifice and the altitude of the fluid above it. Most writers on 

 this subject have considered the column of fluid over the orifice as the expelling 

 force ; whence some have deduced the velocity at the orifice to be that which a 

 body would acquire in falling down the whole depth of the fluid ; and others 

 that acquired in falling through half the depth, without any regard to the mag- 

 nitude of the orifice; whereas it is manifest from experiment, that the velocity 

 at the orifice, the depth of the fluid being the same, depends on the proportion 

 which the magnitude of the orifice bears to the magnitude of the bottom of th 

 vessel, supposing, for instance, the vessel to be a cylinder standing on its base; 

 and in all cases the velocity, caeteris paribus, will depend on the ratio between 

 the magnitude of the orifice and that of the surface of the fluid. Conclusions, 

 thus contrary to matter of fact show, either that the principle assumed is not true, 

 or that the deductions from it are not applicable to the present case. The most 

 celebrated theories on this subject are those of D. Bernouilli and M. D'Alem- 

 .bert; the former deduced his conclusions from the principle of the conservalio 

 virium vivarum, or as he calls it, the equalitas inter descensum actualem ascen- 

 sumque poteniialem, where, by the descensus actualis he means the actual de- 

 scent of the centre of gravity, and by the ascensus potentialis, he means the as- 

 cent of the centre of gravity, if the fluid which flows out could have its motion 

 directed upwards; and the latter from the principle of the equilibrium of the 

 fluid. This principle of M. D'Alembert leads immediately to that assumed by 

 D. Bernouilli, and consequently they both deduce the same fluxional equation, 

 the fluent of which expresses the relation between the velocity of the fluid at 

 the orifice, and the perpendicular altitude of the fluid above it. How far the 

 principles here assumed can be applied in our reasoning on fluids, can only be 

 determined by comparing the conclusions deduced from them with experiments. 

 The fluxional equation above-mentioned cannot in general be integrated, 

 and therefore the relation between the velocity of the fluid at the orifice and its 

 depth cannot from thence be determined in all cases. If the magnitude of the 

 orifice be indefinitely less than that of the surface of the fluid, the equation 



