336 PHILOSOPHICAL TRANSACTIONS. [aNNO I/IS. 



lowing observation, that what I call a periphery, or section of the concave sur- 

 face of the tube, is really a small surface, whose base is that periphery, and 

 whose lieight is the distance to which the attractive power of the glass is 

 extended. 



Of the Motion of Running JVater. By the same Author^ Dr. Jurin, N° 355, 

 p. 748. Translated from the Latin. 



We often see, both in hydraulics and in applying its principles to the animal 

 economy, that the motion of water running through a hole in the bottom of a 

 vessel, is compared with other powers. And since no one has hitherto truly 

 determined the quantity of this motion, hydraulic writers use to assume instead 

 of it, the weight of a column of water incumbent on the said hole. But such 

 as do this, never consider, that it is utterly impossible to compare any motion 

 with a weight that is in a state of rest. But the motion of running water may 

 be easily defined in the following manner. 



Let shahs, fig. 18, pi. 8, be an infinite surface of water, cc a circular hole 

 made in the bottom; ab a perpendicular drawn through the centre of the hole; 

 SGCCGS a column or cataract of water running through the hole cc ; sgc a curve, 

 by whose rotation round the axis ab is generated the solid or cataract sgccgs. 

 For, when water descends freely, and with an accelerated motion, as all heavy 

 bodies do, it is necessarily contracted into a less bulk, as it requires a greater 

 velocity in falling, and runs out at the hole cc with that velocity, which it ac- 

 quires by falling from the height ab. 



But the velocity acquired by a heavy body in its fall, from what Galilaeo has 

 demonstrated, is in a subduplicate ratio of the height from which it has fallen. 

 Therefore, if any ordinate de be drawn to the curve sgc ; and de be called y 

 and AD, Xy the velocity of the water in the section ee will be expressed by y/x^ 

 and the product of that velocity into the section by y/x X y^. But this product 

 is as the mass of water that passes through that section in a given space of time; 

 and since the same bulk of water passes through each section of the cataract in 

 a given time; therefore that product will be always invariable, and s/ x X y^ 

 will be = 1, and ocy'^ = 1. And this is the equation of the curve sgc, whose 

 part contained within the given vessel, was delineated, and its equation not 

 obscurely hinted at by Sir Isaac Newton, in prop. 36, lib. '2, Princip. who was 

 .he first that communicated to the learned world the true velocity of running 

 water, and deduced it from its genuine principles. 



But this curve is an hyperbola of the fourth order, one of whose asymptotes 

 is the right line as, parallel to the horizon, and the other ab perpendicular to 



