VOL. XXX.] PHILOSOPHICAL TRANSACTIONS. 33/ 



it. Its power is the quadrato-cube of the ordinate fg, drawn to the point g, 

 where the right line ag, bisecting the angle comprehended between the two 

 asymptotes meets the curve. The space sades, contained between the curve 

 sge, the ordinate de, and the asymptotes ad, as, is equal to 4- of the rectangle 

 HD, under the abscissa ad and the ordinate de ; and consequently the space 

 SHE is -^ of the same rectangle. And the solid sgeegs, generated by the 

 rotation of the space sades, about the axis ad, is double the cylinder on the 

 section ee; whence the concave solid generated by the conversion of the space 

 SHEGS, about the same axis, is equal to the said cylinder. All which are dis- 

 covered by an easy calculation by the inverse method effluxions. 



Theorem 1 . — If water run out at a round hole, in the bottom of a vessel of 

 an infinite capacity, the motion of the whole cataract of water towards the 

 horizon, is equal to the motion of a cylinder of water whose base is the hole 

 itself, and its height equal to that of the water, whose velocity may equal the 

 velocity of the water running out at the hole ; or is equal to the motion of the 

 bulk of water, that runs out in any given time, whose velocity may be such, 

 that a space equal to the height of the water may be run over in the same 

 given time. 



Demonstration of the first part. — Let another ordinate de be drawn to the 

 curve SGC, very near the former ordinate de. By the rotation of the curve 

 about the axis ab, the ordinates de, de will generate two circles, by which the 

 nascent solid EEee is intercepted. That solid is equal to the product of the 

 height Yid into the section ee ; and its motion is equal to the product of the 

 solid itself into its velocity, or to the product of the height nd, the section ee, 

 and the velocity of the water in that section. And since it has been shown 

 above, that the product of any section of a cataract, and of the velocity of the 

 water in that section, is an invariable quantity ; consequently the motion of the 

 whole cataract, will be equal to the product of that invariable quantity into the 

 sum of all the heights dg^, or into ab, that is, equal to the motion of the 

 cylinder on the hole and the height of the water, whose velocity is equal to 

 that of the water running out at the hole. q. e. d. 



Corol. 1 . — Having given the height of the water, the motion of the cataract 

 will be in the ratio of the hole. 



Cor. 2. — Having given the hole, the motion of the cataract will be in the 

 sesquiplicate ratio of the height, or in the triplicate ratio of the velocity with 

 which the water runs out at the hole. 



Cor. 3. — Having given the motion of the cataract, the hole will be recipro- 

 cally in the sesquiplicate ratio of the height, or reciprocally in the triplicate ratio 

 of the velocity. 



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