302 PHILOSOPHICAL TRANSACTtONS. [aNNO J 789. 



viz, the Measure of the Effluent Water, the Velocity in the Axis of the Con- 

 tracted Fein, or the Altitude to which the Middle Part of the Vein can jet up- 

 wards ; required to determine the rest. 



Let A be the height of the vessel, r the radius of the hole, 2qmr'^A the mea- 

 sure of the effluent water, v the velocity in the axis of the contracted vein, a 

 the height to which the water can jet ; and first let 2qmr^A be given, whence q 

 is given. 



Now, by cor. 2 prob. 10, — = v^3 ; hence v = qvi/3, and u* = a^^v* 



But v^ ?;^: A : a = ^ = S^'a. 



Secondly, if v be given ; 

 then 9 = - /+, and 2qmr^A = —^^ /i, also a = -^. 



Lastly, if « be given ; since a = S^^A, 

 hence q^ =^i and q = v/^— . Also v* = — , and v = vy'-. a. e. i. 



^ O A O A A A 



Prob. 12. Having given the Height to which Water jets through the Air, 

 from a Vessel of a Given Height, through a Given Circular Hole ; to determine 

 the Height to which it will jet from any other Vessel, of any Given Height, and 

 through any Given Circular Hole. 



Let the letters r, s, a, e, q, p denote the same things as in prob. Q ; and let 

 a and e be the heights to which water can jet, when spouting from vessels of 

 the respective altitudes a and e. 



Then, by prob. 1 1, a = S^^a, e = 3/)^e ; hence 

 39^ = \, and 1 - 3^^ = ^, and q = -/^, and;, = -/^, and jb^ = ■^. 



And since, by prob. 9, /> = V\ + C-^~''^y - ^-g^''^' °^ ^y putting 

 ra = nsA, p = Vj + C "g/^ ")" ~ ^ ~6g'' ^ ' ^^"<^^ ^7 substituting ^^-^ for 



1 — 3q^, and y/ j2 for Q> ^"d for a — a writing «, it is /> = ^vali ' 



and ;>' = ^^-^ g^^ • But /> = 5^ ; hence e = 



2Aa + n'>'-««^^4Aa + »'x^ Hence, writing . for e - e, it is ^ = ;^- X 



2ao » 6 2Aa 



(/4Aa + nV — n a). Now, t or e — e bring given, e is also given, or the 

 height to which the water can spout from the new vessel, a. e. i. 



Corol. 1. If the holes in both vessels be equal, or * = r ; then e = nA, 



or n = -, hence £ = ^ X {^ 4Aa + n'«' — nx). 



