YOL. XLI.] PHILOSOPHICAL TRANSACTIONS. 285 



will only descend with the least relative velocity, with regard to the particle a; 

 because otherwise it would carry the particle a. away with it, by accelerating it, 

 and this particle a, being now in a quicker motion, would carry away with it 

 the particle a. In like manner the particle y being placed more within, and 

 contiguous to the particle (3, will descend with the least relative velocity with 

 regard to the particle |3 ; and the other particles S, i, &c. being placed one more 

 within than another, will descend with the least relative velocity, with regard 

 to each of the particles lying next to each of them without. And by this 

 means the absolute velocity of the particles must necessarily increase gradually 

 from the bound towards the centre c, that the velocity of the water may be 

 greatest in the very centre, and least at each bound k and h. 



But it is necessary that the resistance which each quicker particle finds from 

 the friction of the adjacent slower one, placed without, should be perpetu- 

 ally equal through the whole section of the cataract. Otherwise that particle 

 which finds the greater resistance, will accelerate the adjacent slower particle, 

 till the resistance is by this means diminished, and becomes equal to that resist- 

 ance which is found by the other particles. But if the resistance be every 

 where equal through the whole section of the cataract, the relative velocity of 

 the particles will be also equal every where, when one of them necessarily fol- 

 lows another. 



Therefore the absolute velocity of every particle, which is the sum of all the 

 relative velocities, from the circumference of the section to that very particle, 

 taken all together, is in the ratio of the distance of the same particle from the 

 circumference of the cataract. 



Now let r be the radius of the hole, m to 1 in the proportion of the circum- 

 ference to the diameter, mr^ the area of the whole, v the velocity with which 

 the water descends in the centre of the hole, a the height by falling from which 

 in vacuo the velocity v is acquired, a the height of the water above the hole, v 

 the velocity acquired by falling in vacuo.from the height a, t the time of fall- 

 ing from the same, z the distance of any particle from the centre of the hole, 

 and let the water run out in the time t. 



Now the measure of the water, which goes out of the hole in the time t, 

 will be found after this manner: z will be the radius of any circle within the 

 hole, 2mz its circumference, 2mzz the nascent annulus adjacent to that circum- 

 ference, and X V the velocity of the water in that annulus. 



Since v : — u :: 2a : 2ai' X the length of the stream flowing through 



the nascent annulus in the time x; the measure of that water will be 2mzz 



