VOL. XL!.] PHILOSOPHICAL TRAi^SACTIONS. •2€7 



But, from what has been said above., the measure of the water issuing by the 

 hole HK, in the given time t, when the vessel is shortened, is equal to the 

 measure of the fluid passing in the same time through the section hk, when 

 the vessel is entire, or equal to the measure of it issuing by the hole ep in the 

 same time. Iheretore — -|- X — ;— = . ■, or y^^x= r v/a, or xy* = Ar , 

 which is the same equation of the hyperbolical curve, by the rotation of which 

 he formerly showed that the figure of the cataract was generated. 



Scho/.'l. — The measure of the water now found, 2mAr^\/^, or ImAr^ X 

 0.57735, rather exceeds the measure Imxr^ X 0.571, obtained from Poleni's 

 experiments. But this difference, in some measure, arises from not considering 

 the decrease in the motion of the water from resistance, in this problem. 



Schoi. 1. — The measure of the effluent water, as determined by this solution, 

 is accurate, if the height of the vessel be considered as infinitely greater than 

 the diameter of the hole. But as this height has a finite ratio to the diameter 

 of the hole, the measure will be something less, so that, when the height is 5 

 times greater than the diameter, it will diff^er from the truth only the 320(X)th 

 part, and when it is double, only about the 5 1 20th part, which differences are 

 smaller than can be discovered by any experiment. And this small difference 

 proceeds from hence, that the abovementioned relative velocity, and therefore 

 the absolute velocity of the particles of water, which have been considered as 

 in a direction perpendicular to the horizon, are really in a direction somewhat 

 oblique, when every particle comes near the axis of the cataract in descending. 



But if a true and accurate solution be desired, when the altitude of the 

 water has any ratio whatever to the diameter of the hole, it may be done as 

 follows. 



From the property of the cataract curve, in corol. 2 of this problem, viz. 

 xy* = Ar*. the subtangent of this curve at the place of the hole will be 4a, and 

 at the place of any section the subtangent will be 4x, that is, 4 times the 

 height of the water above that section. But such a cataract curve is described 

 not only by the exterior water, which flows beyond the hole, but also by that 

 part of the water which flows through any annulus of the hole, that is, every 

 particle of water describes such a curve. 



Now let 2 be the distance of any particle, in the hole, from 'its centre, and 

 let this particle descend through the smallest space in a tangent to the cataract 

 curve. Hence its velocity in the direction of the tangent, or the velocity 

 — ^^ V, explained in this problem, will be to the velocity in the perpendicular 

 direction, as V^i6a* + z^ is to 4a; therefore the velocity in the perpendicular 

 direction will be -7==== x — - " " ' „. 



