■286 PHILOSOPHICAL TRANSACTIONS. [aNNW I739. 



r—x , ^^ rzx—z^z .,^a . „f .I- 1 ..■ 2OTAD. 



X 2av X = 4mAv X ; the fluent of which, viz. X3?V— 2z', 



vr \r 3vr 



when z = r, gives — - — for the measure of the water passing through all the 

 hole in the time t. 



But the motion of the same water will be also found thus. The measure of 

 the water running through the nascent annulus, in the time t, being 



-^^ X rzz — z'z, and its velocity being v X — — , its motion will be — — X 



4mAt)* 



rzz — z'^i X n X = o- X r'^zz — 2rz^z + z'z, the fluent of which, 



T V/* 



when z = r, gives — — for the motion of the water, running out in the time 

 T, through all the hole. 



But this motion is equal to that which the column over the hole can acquire, 

 in the same time t, by falling by its own weight through a vacuum, that is to 

 the motion afv, or av X mr^; therefore — - — = mavr^. Hence v = v \/ 3. 



Also the abovementioned measure of the water issuing at the hole in the 

 time T, viz.—- - = „ — X vv^3 = —75-. a, e. i, 



' 3v 3v v3 



Carol. 1. — Since v^ : v^ :: a: a, therefore a = ^ = ^ X 3v^ = 3a. Tliere- 

 fore the height to which the water can rise, with that velocity with which it 

 runs out in the centre of the hole, is triple the height of the fluid above 

 the hole. 



Carol. 2. — ^The figure of the cataract will be determined in the following 

 manner : 



Let HK, fig. t), be any section of the cataract, having the centre c; and let 

 its radius ck = y, the height of the water above that section, or ci = x, t the 

 time of falling in vacuo from the height x, and as before let lf = r, and 



LI = A. 



Now the water passes through this section hk in the same quantity as it runs 

 out of the hole ef. But if the vessel be shortened, so that its height be 

 reduced from il to ic, and so that the section now becomes the very hole in the 

 bottom of the vessel, the water will pass through this section in a given time, 

 in the very same quantity as it passed through the same before the vessel was 

 shortened. 



Now the vessel being shortened, the measure of the water issuing by the 

 hole HK, in the time t, by the preceding solution, is — ^ , and the measure of 



the fluid issuing in the time t, is —'^ x j = -j^ X — ^: for t : r ::/A:y^ar. 



i 



