VOL. XLI.] PHILOSOPHICAL TRANSACTIONS. 279 



brought to the test by experiments; and those who laboured in making experi- 

 ments, omitting to observe some minute circumstances, the importance of 

 which they had not yet perceived, differed greatly from one another, and al- 

 most all of them erred from the real measure. 



Of this there cannot be given a better example, than that simple and easy 

 one, which has generally been a foundation for all the rest, and is what we 

 have now undertaken to treat of diligently, when water issues through a circu- 

 lar hole made in the bottom of a vessel constantly full, with a constant velocity, 

 Poleni alone has given the true measure of the water flowing out, or at least 

 very near the true one; and Sir I. Newton alone has laid the foundation of 

 discovering that measure; though most have rejected it, and some, concealing 

 the author's name, have pretended that it was their own. 



We shall therefore make our attempt under the conduct of these two leaders; 

 and in the first place propose, under the name of phaenomena, such things as 

 either appear from experiments, or are confirmed by certain reasonings drawn 

 from them; and in the last place, we shall attempt the solution of those phse- 

 nomena. 



Phenomena of Water Rowing through a Hole in the Bottom of a Vessel 

 constantly full. — 1. The depth of the water, and the time of flowing out being 

 given, the measure of the effluent water is nearly in proportion to the hole. 



1. The depth of the water, and also the hole being given, the measure of 

 the effluent water is in proportion to the time. 



3. The time of flowing out, and the hole being given, the measure of the 

 effluent water is nearly in a subduplicate proportion to the height of the water. 



4. The measure of the effluent water is nearly in a ratio compounded of the 

 proportion of the hole, the proportion of the time, and a subduplicate propor- 

 tion of the depth of the water. 



5. The measure of water flowing out in a given time, is much less than that 

 which is commonly assigned by mathematical theorems. For the velocity of 

 effluent water is commonly supposed to be that which a heavy body would 

 acquire in vacuo by falling from the whole height of the water above the hole, 

 and this being supposed, if we call the area of the hole f, the height of the 

 water above the hole a, the velocity which a heavy body acquires by falling in 

 vacuo from that height v, and the time of falling t, and if the water flows out 

 with this constant velocity v, in the time t; then the length of the column of 

 water, which flows out in that time, will be 2a ; and the measure of it will be 

 2ap. But if we calculate from the most accurate experiments of Poleni, we 

 shall find the quantity of water which flows out in that time, to be no more 

 than about v*oVo of this measure 2af. 



