306 A. B. (^uinby on the Overshot F/ater-Whed. 



arc ADB,) is equal to that of the particle P, estimated for the 

 same time. 



And as the effects produced by the two equal particles, 

 (or powers,) P and P', during any given time, will obviously 

 be to each other, as the mean tendencies of those particles, 

 (during the same time,) to produce rotation ; it follows that 

 the effect produced by the particle P', in descending from A 

 to B, in the arc ADB, will be equal to that produced by the 

 particle P, in the same time; or = WxP?/. 



Hence, if a particle of water descend upon an overshot 

 water-wheel, which is the whole height of the fall, it will 

 raise an eq^al particle through the same vertical height: — •■ 

 and, as this will be the case with every particle, it follows that 

 any quantity whatever of water, descending upon an over- 

 shot water-wheel, which is the whole height of the fall, will 

 raise an equal quantity through the same vertical height. 



The same can be demonstrated in a different manner. 



Let the circle ABE, Fig. 2, represent an overshot wheel ; 

 and let P, P', P", &;c. be different situations of the same par- 

 ticle of water P : and suppose each of the arcs PP', P'P", 

 &.C. to be an indefinitely small element: — then, each element 

 and its chord, (and also its sine and tangent,) may be con- 

 sidered as coinciding. Wherefore, for the value of P, in the 

 respective elements, we have, (by mechanics,) 



PxP/ Px/m PXmft PXno 



p p/ ' pf p ' ' p/j p/^/' p'/ 'pmi ' 



and, for the effect of P in the respective elements, (or, for the 

 tendency which P has in the respective elements to raise the 



PXP/ „„ Fxlm 

 equal particle VV,) we have p p, ' P P', pr~p//' P' P"j 



.. p" p//' " '""" . P"' P"" =PvP/ Pv/rw 



p// -pin ^ ^ 5 p/« p/'" ^ f ■> — XAAt, JT AiWj 



PXtww, Pxno ; whence it is plain that the effect of P, in the 

 respective elements, is, always, as the perpendicular space 

 passed through ; and, also, that the (ffect of P, in descending 

 through any number of elements; or through any portion 

 whatever of the arc P, P', P ', Ssc. is, always, as the perpen- 

 dicular distance passed through ; or, as the vertical space in- 

 tercepted between the point A and a perpendicular drawn 

 from the lower extremity of the last element, to the line AB. 

 Hence, if it can be proved that the effect of P. in any one 



