A. B. i^uinby on the Overshot Water-Wheel. 303 



of its descent, will be sufficient to raise the particle W through 

 an equal space. 



Let it next be considered, what effect the particle P would 

 have, during its descent, through some particular or assumed 

 space. 



Take Py=AB, and it will be manifest that the effect of the 

 particle P, duiing its descent to tlui point y, will be properly 

 expressed by the product PxP?/, or PxAB = WxBA, Let 

 it also be considered, that during the descent of the par- 

 ticle P, from P to I/, the wheel will be made to turn through 

 half a revolution ; for, since by cons. AD ; CD; ;CD : CG ; 

 and by the property of circles AD : CD: :^G t CG, it fol- 

 lows that ^G=CD; and, consequently, Gi!t;=2CD—AB-—P;y: 

 and, therefore, if we suppose a particle of water to be at A 

 when the particle P shall begin to descend, it will have de- 

 scribed the arc ADB, and have arrived at the point B, at the 

 time that the f)article P shall have arrived at t/. 



It is now proposed to estimate the effect which a particle 

 of water P',=to P, would have in descending from A, through 

 the arc ADB ; in comparison with the effect that would be 

 produced, during the same time, by the particle P, acting up- 

 on the teeth of the wheel, at the point G. 



From P' let fall the perpendicular P'n; and it is manifest 

 that the tendency which the particle P' has to produce rota- 

 tion, is to that which the particle P has to produce rotation, . 

 in the ratio of Cn to CG. If, therefore, CG be taken to ex- 

 press the tendency of the particle P to produce rotation; then 

 that of the particle P' to produce rotation, will be properly 

 expressed by the line Cn; the perpendicular distance of the 

 particle P' from the line ACB. And, in general, the ten- 

 dency of the particle P' to produce rotation, at any point 

 whatever of the semicircle ADB, will be expressed by the 

 perpendicular distance of that point from the line ACB. 



Hence, to determine the mean tendency of the particle P' 

 to produce rotation, (m terms of CG,) during its descent from 

 A to B, in the arc ABD, we must find the mean distance of 

 the semicircular arc ADB from the line ACB; which, by 



CD2 CD2 



Vince's Flux. p. 97, is=-Tj^ — ; but, CG was made =TT) — J 



and, therefore, the mean tendency of the particle P' to pro- 

 duce rotation, during its descent, (from A to B, through the 

 VoLF IX.— No. 9. 39 



