154 THIRD REPORT — 1833. 



particles of the water ; from which he concluded that the law 

 of acceleration existed, and that the particles which escaped at 

 every instant of time received their motion simply from the 

 pressure produced by the weight of the fluid column above the 

 orifice, and that the weight of this column of fluid ought to 

 represent the pressure on the particles which continually escape 

 from the orifice ; and that the quantity of motion or expenditure 

 is in the ratio of the breadth of the orifice, multiplied by the 

 square of the velocity, or, in other words, that the height of 

 the water in the vessel is proportional to the square of the ve- 

 locity with which it escapes ; which is precisely the theorem of 

 Torricelli. This mode of reasoning is in some degree vague, 

 because it supposes that the small mass which escapes from 

 the vessel at each instant of time acquires its velocity from the 

 pressure of the column immediately above the orifice. But 

 supposing, as is natural, that the weight of the cohunn acts on 

 the particle during the time it takes to issue from the vessel, it 

 is clear that this particle will receive an accelerated motion, 

 whose quantity in a given time will be proportional to the 

 pressure multiplied by the time : hence the product of the 

 weight of the column by the time of its issuing from the orifice, 

 will be equal to the product of the mass of this particle by the 

 velocity it will have acquired ; and as the mass is the product 

 of the opening of the orifice, by the small space which the 

 particle describes in issuing from the orifice, it follows that the 

 height of the column will be as the square of the velocity ac- 

 quired. This theory is the more correct the more the fluid 

 approaches to a perfect state of repose, and the more the 

 dimensions of the vessel exceed the dimensions of the orifice. 

 By a contrary mode of reasoning this theory became insufficient 

 to determine the motions of fluids through pipes of small dia- 

 meters. It is necessary, therefore, to consider all the motions 

 of the particles of fluids, and examine how they are changed 

 and altered by the figure of the conduit. But experiment teaches 

 us that when a pipe has a different direction from the vertical 

 one, the different horizontal sections of the fluid preserve their 

 parallelism, the sections following taking the place of the pre- 

 ceding ones, and so on ; from which it follows (on account of 

 the incompressibility of the fluid) that the velocity of each 

 horizontal section or plate, taken vertically, ought to be in 

 the inverse ratio of the diameter of the section. It suffices, 

 therefore, to determine the motion of a single section, and the 

 problem then becomes analogous to the vibration of a com- 

 pound pendulum, by which, according to the theory of James 

 Bernoulli, the motions acquired and lost at each instant of time 



