NEWTON S PRINCIPIA. 381 



of the section of the vessel that the stratum is passing. 

 He then applied the principle of the conservation of vis 

 viva to determine the velocity of efflux. Bernoulli was 

 led to this hypothesis by observing the manner in which 

 particles of Spanish wax immersed in the water moved 

 along with the fluid. A theory thus founded on obser 

 vation is usually a great step in advance. Even now, in 

 certain cases, we are obliged to have recourse to this very 

 assumption. John Bernoulli (Hydraulique) gave a dif 

 ferent theory. Taking a stratum of the fluid, he replaced 

 the force which would produce its motion by another 

 supposed to act at the surface of the fluid. Then by in 

 tegration he obtained the whole force that acting at the 

 surface would produce the whole motion of the fluid, and 

 this force he assumes to be equal to the whole weight of 

 the fluid. There would now be no advantage in dwelling 

 on these or any similar investigations. By the aid of 

 D Alembert s principle we have now correct equations, 

 giving the motion of fluids under all circumstances. We 

 shall therefore pass on to the more satisfactory solutions 

 that Poisson ( Traite de Mecanique, torn. ii. chap, iii.) 

 gave of the question. 



(4.) In modern times, it is usual to deduce the velocity 

 of efflux from the equations of motions. The simplest 

 case is when the motion is steady. Let the surface of the 

 fluid be always horizontal, and retained at the same level ; 

 let the area be A, and suppose all its parts to have the 

 same velocity (U) in a direction vertically downwards. 

 Let B be the area of the section of the fluid at the ve?ia 

 contract^ V the velocity of the fluid at that point. A 

 complete solution of the question ought to determine both 

 B and V in terms of the diameter of the hole and the 

 other circumstances of the problem. According to the 

 ordinary notation we have, 



