288 PHILOSOPHICAL TRANSACTIONS. [aNNO 1739. 



And hence, after the manner of the above solution, the measure of the 

 water passing through the nascent annulus will be — ^ X —p==L=z. The 



fluent of which being taken, either by Cotes's forms or by infinite series, when 

 properly 'corrected, will give the whole quantity run out by the hole in the time 

 t; which in a series is the quantity ?^ X 1 - ■—--, + Jq^, — &c. 

 Hence, by supposing a to be infinitely greater than r, or the height than the 

 hole, the measure comes out barely 2A.mr^\/-y, the same as was deternjined be- 

 fore. Hence also. 



When A = lOr, the measure is ^Amr^v^-r X (1 — ttttto) nearly. 



And when a = 4r, it is 2AOTr^'/4- X (1 — -nVo) nearly. 



So that, instead of the true measure, we may always take 2AmT^\/-^, without 

 any sensible error, even in so small an altitude, and much more in an altitude 

 many times greater, as it usually is in experiments; which makes the computa- 

 tion very easy, 



Prob. III. — Supposing again the same thing as before, and neglecting the acce- 

 leration of the water without the hole; required to determine the Diameter oj" the 

 F^ein of water at the small distance without the hole, where the vein is most con- 

 tracted, and the Velocity of the water in the Vein so contracted. 



In the solution of the former problem it was observed, that the particles of 

 water passing through the hole, do not all issue with the same velocity, but 

 every one with a greater velocity as it is nearer the centre ; and that the relative 

 velocity of the inner particles, with respect to the particles that touch each of 

 them on the outside, is constantly equal through all the hole ; and this relative 

 velocity proceeds from the resistance given to the particles, by the surrounding 

 water, as they descend towards the hole. 



But after the water has passed the hole, and its outer surface is no longer 

 resisted by the surrounding fluid, nor by the ambient air, because moving in a 

 vacuum by the hypothesis, that relative velocity, or inequality of absolute velo- 

 city, can no longer obtain. For now the swifter particles must necessarily acce- 

 lerate the slower contiguous ones, and must also themselves be retarded by the 

 slower, till all the particles have acquired one common velocity, which will 

 happen at a small distance without the hole. 



But while all the particles are acquiring this common velocity, the diameter 

 of the vein must necessarily be contracting. This happens in the same manner, 

 as when a rapid river is joined with a slower, as the Rhone with the Saone: in 

 the common channel, the velocity of the water from both rivers is equal, and 

 the water passes through a section of this channel in like quantity as before, 

 through the sections of both rivers ; though a section of the Rhone below the 



