VOL. XXX.] PHILOSOPHICAL TRANSACTIONS. 325 



Here it may not be improper to take notice, that the demonstrations of those 

 three learned persons are all grounded on this supposition, that the water 

 running out from the hole is pressed on by the column of water incum- 

 bent on it, which may easily be demonstrated to be a mistake. Likewise, if 

 their demonstrations be just, it will follow, that the first drops of water, 

 which issue out from the hole, must run with the same velocity, as after the 

 water has been running some time ; the contrary of which appears to be true 

 in fact by the experiments of M. Mariotte. 



The author might have found a juster account of this matter in the 

 writings of a great man, (Newton) whom he cites on another occasion ; the 

 second edition of whose book was come out some time before Herman 

 published either of those demonstrations, and had been seen by him, as appears 

 by his quoting it frequently, and mentioning the difference in this very parti- 

 cular between the first and second edition. 



Our author goes on to consider the simple motion of water running out by 

 a section perpendicular to the horizon, in the side of a receptacle, which is 

 always kept up at the same height. He shows, that the velocities with which 

 the water issues out at different depths, being as the roots of those respective 

 depths, may be represented by the ordinates of a parabola, whose axis repre- 

 sents the entire depth of the water. Consequently, since the quantities of 

 water, running out at different depths, are as those velocities, they likewise 

 may be represented by the same ordinates, and the quantity of water discharged 

 from the whole section, will be represented by the parabolic space; and the 

 mean velocity by that same space divided by the abscisse. 



The author proceeds next to the mixed motion of water ; in order to dis- 

 cover the nature of which, he has made some curious experiments. 



In order therefore to find a general rule for determining the proportion be- 

 tween the parabolic spaces, which represent the quantities discharged by the 

 mixed and simple motion, or between the parameters of those parabolas, he 

 draws some observations from the foregoing experiments, by the help of which 

 he hopes such a rule may be found out. 



In effect the author proceeds, in a tentative method, to find his rule, and 

 having discovered it, he shows by calculation, that it answers all the conditions 

 before required. This rule is expressed in a pretty high equation; which, 

 besides other operations, requires the extracting the root of the sixth power. 

 From this equation is derived another, serving to find either the quantity of 

 water discharged, the depth of the running, or that of the dead water, the 

 other two of them being given; as likewise a third equation, to find the mean 

 velocity. 



