VOL. XXX.] PHILOSOPHICAL TRANSACTIONS. 339 



hole, and height ab, whose velocity is equal to that of the water running out 

 at the hole. a. e. d. 



The demonstration of the second part follows from the first. 



Cor. ]. — Hence, by substituting the height ab, instead of that of the water, 

 there arise all the corollaries of the preceding proposition. 



Cor. 1. — If the vessel be of any other figure than cylindrical ; or the figure 

 of the hole be square, triangular, or any other except round; or if the hole it- 

 self be not in the middle of the bottom ; or if it be made even in the side of the 

 vessel ; still the motion of the cataract will be the same, viz. equal to the mo- 

 tion of the prism of water on the hole, and height ab whose velocity is equal 

 to that of the effluent water. For the same bulk of water will pass with the 

 same velocity, as in the former hypothesis, both through the hole itself, and 

 through each section of the cataract. 



Cor. 3. — If the diameter of the vessel have a very great ratio to that of the 

 hole, the height ad may be neglected, and the height of the vessel itself may 

 be assumed for that of the cylinder or prism of water. 



Hitherto I have considered apart, that particular case in which water, by the 

 force of gravity, runs out at a vessel. And the rather, both because mathe- 

 maticians mostly apply only this in treating of the impetus of fluids, and be- 

 cause I think the above explained property of the hyperbolic curve, by which 

 the cataract of falling water is formed, not unworthy the consideration of geo- 

 metricians. Otherwise that case might be easily deduced from the following 

 general theorem. 



Theor. 3. — If water run through any full pipe abcd, fig. 20, in the direction 

 of the right line ef^ to which both the orifices of the pipe, ab and CD are per- 

 pendicular ; the motion of the water towards the orifice cd, or the motion of 

 the impediment, which in that orifice opposes the motion of the whole water, 

 is equal to the motion of the prism of water on any section of the pipe gh, 

 and the line of direction or the length of the pipe ef, which moves with the 

 same velocity with which the water flows through that section : or is equal to 

 the motion of a bulk of water, which runs out at the pipe in any given time, 

 and whose velocity is such that a space equal to the length of the pipe is run 

 over in the same given time. 



Case 1. — Let the line of direction be any right line ep. 



The first part is easily demonstrated, in the same manner as Theor, 1 . For, 

 the product of any section of the pipe gh, and of the velocity of the water in 

 that section, is an invariable quantity. 



The second part follows from the first. 



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