2Q2 PHILOSOPHICAL TRANSACTIONS. [aNNO 173Q. 



duct of the relative velocity, and of the density of the accelerating particle of 

 water, be equal to the product of the relative velocity, and of the density of 

 the retarding particle of air. But the density of air is to the density of water 

 as 1 to 900 nearly. Therefore the relative velocity between the outer particle 

 of water and the next of air, is to the relative velocity of the two next particles 

 of water, as 90O to I nearly, 



Also, that inmost particle of^ air is solicited by the next contiguous particle 

 of water to accelerate the motion, and retarded by the next particle of air out- 

 wards. And as here two contrary forces are equal to each other, the product 

 of the relative velocity and density of the accelerating particle of water, will be 

 equal to the product of the relative velocity and density of the retarding particle 

 of air. Therefore the relative velocity between those two particles of air, will 

 be to the relative velocity between the inmost particle of air and the next of 

 water, as gOO to 1 nearly ; and it will be to the relative velocity between the two 

 next particles of water, as QOO X QOO to 1 nearly. And this great relative 

 velocity will be always the same through the whole thickness of the ring of 

 air, which is drawn into motion by the effluent water. 



Now let r, m, v, a, v, a, t denote the same things as in prob. 2. Also let 

 V be the velocity of the water in the axis of the contracted vein, p the radius 

 of the same vein, and r the radius of an imaginary vein, by which the velocity 

 u, by decreasing gradually, in like manner as it decreases in the true vein, is 

 at length reduced to nothing. Also let the measure of the water passing 

 through the hole in the time t, be Iqmr^K. 



Now the measure of the water running in the contracted vein, in the same 



time, by proceeding as in Prob. 2, will be — — - X 3r — 2p. 



But these two measures are equal ; therefore Zqr'^ rv = up^ (3r — 2p). , 



Further, as the measure of the water running through the hole in the time 

 T, is Iqmr'^A, the motion of the same, by Prob. 6, is 3q^mr^Av. And the 

 motion of the water running through the vein in the same time, by proceed- 

 ing as in Prob. 2, is found ^^, X (6rV — 8Rp^ -|- 3p'). 



Now these two art equal; and hence Q^V^rV = u^(6rV — 8Rp^ -|- 3p*). 



Then these two equations being rightly reduced for exterminating r, we 

 come to the following equation, p^u'^ + 1q\jvr^f^ = I2q\'^r^f —Qqv'^r*. From 

 which may be found any of the three quantities p, u, 5 ; viz. 

 g = ^ ^/^v X \/(u -I- 69V — 2v/(3^uv -f 9^V — 2u^), 



,=:^-^X (,- + 2^/3p'^-2r^), 



g'" 



