VOL. XLI.] PHILOSOPHICAL TRANSACTIONS. 289 



junction is much less than the sum of the two sections of both rivers above 

 the same. 



Therefore let the radius of the contracted vein of water, where all the par- 

 ticles in its section have acquired an equal velocity, be f, and let that common 

 velocity be called u: then as v : y :: 2a : — the length of the vein, and there- 

 fore — " X m^^ Js the measure of the water passing through the section in the 



time t; the motion of which in that time is therefore — . 



But the measure of the water passing through the section of the vein, must 

 be equal to that passing through the hole in the same time, that is, 



— V— = -::^'°'^?"-^' 



Also the motion of the water through the hole, as it is not altered by the 

 action of the particles on each other, must be equal to the motion of the water 



through the section of the vein, that is, Avmr'^ = — , or 2f^u^ = r^v'^. 



Hence, dividing this equation by that immediately above, it gives 



Hence j" = ~ ='^x ■^, = ^r\ and j = r/^. q. e. i. 



Corol. — Since v^ = 4v*, and the altitudes are in the duplicate ratio of the 

 velocities generated by falling through them, therefore this is the velocity of 

 the water in the contracted vein, by which it can jet upwards in vacuo to ^ of 

 the height of the fluid above the hole. 



Scholium. — This extraordinary contraction of the vein of water was first dis- 

 covered about 30 years before, by Sir I. Newton, when he was considering the 

 motion of effluent water more attentively, on account of some difficulties pro- 

 posed by Mr. Cotes, who was then taking care of the 2d edition of the Prin- 

 cipia; and Poleni afterwards confirmed it by many experiments. From that 

 time this phenomenon has greatly exercised the wits of philosophers, without 

 however detecting the true cause of it. 



The radius of the vein, ry^^, orO.Sldor, determined by this problem, is a 

 little less than 0.84r, as delivered by Sir Isaac; and a little greater than 0.78r, 

 according to Poleni's measure, being indeed nearly a mean between them both. 



Prob. IV. — Having given the measure of effluent water, through a circular 

 hole in the bottom of a cylindrical vessel; to determine the motion of the same, 

 and the velocity in the centre of the hole. 



Let the given measure of the water, issuing in the time t, be 1mr\q; to 

 which the measure assigned by the analysis in prob. 2 will be equal, viz. 



vol. VIII. P p 



