ago PHILOSOPHICAL TRAKSACTIONS. [aNNO 1739- 



Imr^A.q = — - — , or t; = 3vq. But the motion of the same water, assigned 



by the analysis in the same problem, is — — ; and by substituting here, instead 

 of tJ^, its value just now found, that motion becomes Sq'^mr^AV, a. e. i. 



Carol. — If from the motion which can be generated in the time t, by the 

 column of water over the hole, viz. mr^Av, be subtracted the motion of the 

 water running out in the same time, viz. Zq'^mr^Av, there remains mr'^AV X 1 — 3q'^ 

 for the motion lost by the resistance in the time t. 



Prob. V. — fVlth the same data and suppositions as before, and neglecting the 

 acceleration of the water without the hole; it is proposed to determine the dia- 

 meter of the vein of water at a small distance without the hole, where the vein is 

 most contracted, and also the velocity of the water in the vein so contracted. 



By prob. 3, the measure of the water passing through a section of the vein, 



in the time T, is — - — ^; which is also equal to the given measure Imr^Aq; 

 hence f'^u = r'^vq. 



Again, by prob. 3, the motion of the water passing through a section of the 

 vein, in the time t, is — - — , to which is equal the motion determined by the 

 former problem, viz. Sq^mrlAV; which gives 2fV = S^Vv^. But 

 M = -^ — -^-7- = -l^v; and j'' = — i = r^vo X rr- = t'' ; therefore 



= r\/^. Q. E. I. 



Corol. 1 . — The same ratio remains between the radius of the hole and that 

 of the contracted vein, whether the motion of the effluent water be any how 

 diminished by resistance, as in this problem, or not diminished as in prob. 3, 

 being both ways the same, j = r\/-^. 



Corol. 1. — When the motion of the effluent water is diminished by resistance, 

 the velocity is at the same time diminished in the contracted vein. For in prob. 

 3 it was u = -^v\/3, but now it becomes u = -l^v; so that u is diminished from 

 0.866v to 0.856v, taking q = 0.571, according to Poleni's experiments. 



Prob. VI. — Supposing the water issuing through a circular hole in the middle of 

 the bottom of a cylindrical vessel, when the particles of water, as they floiv down- 

 wards within the vessel, suffer so great a resistance from a want of lubricity, that 

 the motion of the fluid is much diminished by it, and also the measure of the ef- 

 fluent water being given; it is proposed to determine the motion of the same, and 

 the velocity with which it passes through the middle of the hole. 



Let the given measure of the water, issuing in the time t, be Imr'^Aq, as in 

 prob. 4 ; and by that prob. we have for the motion of the same Sq'^mr^Av, also 

 the velocity with which it passes through the centre of the hole, or ti = 39V. 

 a. E. I. 



