VOL, 



XLI.^ PHILOSOPHICAL TRANSACTIONS. igg 



it becomes equal toy ; then the fluent of the above expression will be — — i-, or 



^!^i', by making z = i/, which will be as the resistance of the nascent cylinder, 

 of the radius 1/ and altitude x. 



But, by the nature of the cataract curve, xy* = Ar*, ory^x = r^A; hence 

 the resistance of this nascent cylinder will be as ^^.r ; and the resistance of 



the whole cataract, will be as the fluent of this fluxion, or as ^mvr^-, or as 



^mvr^ ti? by making ar = a. And since, by prob. 4, it is v = S^v, the re- 

 sistance in the cataract will be as ^^mvrV a?, or qxrV x^. a. e. i. 



CoroL- — Since v is as v'a, the resistance in the cataract will be as qiA?. 



Schol. — In the above solution, Imzi' has been used instead of 2mzV.f^ +^j 

 the true quantity ; and if this be used, as also the subtangent and tangent of 

 the curve as in prob. 4 ; then, by going through the same process as above, 

 the resistance through the whole cataract will be as 



irmvry/A? X I —775^71 + W:\s:Va* ~ sStW + ^^ -^"^ '^ ^'^^ altitude a 

 be considered as infinite, with respect to the diameter of the hole, all the terms 

 of the series after the first will vanish, and the resistance will be barely 

 as ^mvr^A?, the same as before determined. 



If A = lOr, the resistance will be as ^mvrt/ a^ X (1 7-) nearly. 



If a = Ar, the resistance will be as \mvr*/ a^ X (1 5) nearly. 



We may therefore use \mvr^ a^ for the measure of the resistance, without 

 sensible error, even where the altitude of the water does not exceed 2 diameters 

 of the hole, and much more in a far greater height. 



Prob. 9. — Having Given the Measure of the Water issuing through a Given 

 Circular Hole, in the Middle of the Bottom of a Circular f^essel, of a Given 

 Depth ; to determine the Measure of the Water issuing from another Vessel of 

 Any Given Depth, through Any Given Circular Hole. 



Let r denote the radius of the given hole, a the altitude or depth, 2mqr^A 

 the given measure of the water issuing in that time in which a body would fall 

 in vacuo through the altitude a. 



Then, by prob. 4, S^^mr^AV will be the motion of the water issuing in the 

 same time ; and, by the cor. to prob. 4, the motion lost in the same time by 

 the resistance will be mr^Av(i — 3^^). Hence therefore an equal force of re- 

 sistance can generate this motion in the same time. 



But the motions generated in the same time are proportional to the generat- 

 ing forces. Therefore the motion mr^AV, which the weight of the column of 

 water mr* a can generate in this time by prob, 1, without any resistance, is to 

 the motion mr^Av(l — 39^), which the resistance can generate in the same 



ao 2 



