284 PHILOSOPHICAL TRANSACTIONS. [aNNO 1739. 



hole, and therefore not to be generated but by double this column, as in the 

 Princip. ed. 1 and 3, lib. 3, pr. 36. 



Scholium. — ^This measure here determined kf^I, or 2af X O.707, as it 

 falls far short of that which is generally determined by mathematicians, viz. 

 2af, so it far exceeds that measure which is shown by Poleni's experiments, 

 or 2af X 0.57 1 ; and no wonder, for what is supposed in this problem, that 

 the particles of water find no resistance in running down, the hypothesis is far 

 from the true state of things. 



Prob. II. — To determine the Motion, Measure, and J^elocity of water, running 

 out into a Vacuum, through a Circular Hole in the middle part of the bottom of a 

 cylindrical vessel, where the particles of water find Some Resistance for want of 

 a Lubricity, but so small that the decrease of the motion of the effiuent water occa- 

 sioned, cannot be accounted any thing. 



Let abcd, fig. 4, pi. 7, be an immense cylindrical vessel: ef a circular hole 

 made in the middle part of the bottom; and, the water being perfectly at rest 

 and unmoved in the vessel, let the stopper be removed from the hole, that a 

 passage may be opened for the water through it. 



Then because the water was at rest, and now begins to run out through the 

 hole, and the water placed above follows that which runs out, and the natural 

 motion of the water is not disturbed by pouring any over it, and the hole is in 

 the very middle of the bottom, that portion of water which is in motion, and 

 descends towards the hole, will necessarily assume some regular figure ahefkb, 

 of which the lower base is the hole itself, and the upper base, the upper sur- 

 face of the water ab, and all the horizontal sections are circular. This is called 

 a cataract; but we do not yet examine what is the figure of the cataract: it is 

 sufficient for our present design, to observe that it is regular, and that the same 

 quantity of water passes in a given time through each of its horizontal sections. 



Now because all that water which tends downwards, is contained in the cata- 

 ract, it follows that the rest of the water ahec, bkpd, which is without the 

 cataract, has no motion at all, and is perfectly at rest. Therefore in any hori- 

 zontal section of the cataract hck, whose centre is c, the points h, k, shall 

 represent the bounds between the water descending towards the hole, and the 

 surrounding quiescent water. 



Also, as the point k is the bound of motion and rest, and the particles of 

 water, while they are in motion, find a resistance for want of lubricity, the 

 particle of water a. within the cataract, fig. 5, next to the point k, must be 

 carried downwards only with the least velocity. Otherwise it would necessarily 

 carry with it the next particle a, placed without the cataract, contrary to the 

 hypothesis. But the particle (3, which is contiguous within to the particle «, 



