VOL. XLI.] PHILOSOPHICAL TRANSACTIONS. 301 



water jets, and the altitude of the vessel, has nearly the duplicate ratio of the 

 altitude of the vessel. Therefore let a be the height to which the water, in 

 the axis of the vein, with the velocity v, can jet ; then, by Mariotte's experir 

 njents, a — a is as a^ and — — ■ is a given quantity. 



But in one experiment, which Mariotte esteems a fundamental one, a was 

 = 60 Paris inches, and he found a = 5Q inches, the diameter of the hole being 

 half an inch. So that in this case — — — = 3600; and since this is a given quan- 

 tity, it will be always SSoOa = 3600a — a% or a = — ^g^" "^ = A — ^. 



Therefore, if a = 1 inch, or double the diameter of the hole, it will be 

 a= 1 — rir. But v^ :\^ ::a: \: 1 — ^^ : 1 . Therefore when the altitude 

 of the vessel is double the diameter of the hole, there may be taken v^ = v'', 

 or t; = v. 



Further, by cor. 4, prob. Q, as e decreases, p verges to \/-^. Therefore 

 wlfen the altitude of the vessel is very small, as about 2 diameters of the hole, 

 then we may takejb or 9 = -/-J-* , 



But, by prob. 7, j'' = i^ X (v + 6qv — Wzqvv + g^^v'—lli^), and here 

 instead of v and q substituting their values just found, or v and -v/-^, there results 

 p'' = rViX (I +2i/3 -2^1 + ^/3), or f^ = r^ X {I -V V ^ — 2\^T+V\) 

 = 1^ X 0.6687553907 ; and hence p = r X 0.81777466. 



Here then is the value of p, when the altitude of the water is double the 

 diameter of the hole : and since by schol. 2, prob. 7, p obtains a constant ratio 

 to the radius of the hole, it will have the same value in any altitude of the 

 water, q. e. i. 



Carol. 1. By prob. 7, k = ip + /-j— => hence by the value of p just 



found, there arises r = r X 3.98877 150, being the value of r when the altitude 

 of the water is double the diameter of the hole : and since, by schol. 2 of the 

 same problem, the ratio between r and r is constant, therefore r will have this 

 same value whatever the altitude of the water be. 



Carol. 2. Because v is nearly = v, and q nearly = y^J- when the altitude of 

 the water is double the diameter of the hole ; therefore, at this altitude of the 

 water, — = \/3 very nearly. And since, by schol. 2, prob. 7, the ratio between 



V and ^v is constant, therefore — will be = v^3, whatever be the altitude of 

 the water. 



Prob. 1 1 . The Water issuing from a Vessel always full, through a Given 

 Hole, into the Air ; and having Given any one of the three following quantities. 



