VOL. LI.] PHILOSOPHICAL TRANSACTIONS. 341 



and if a power can raise half the weight to double the height; or double the 

 weight to half the height, in the same time that another can, those two powers 

 are equal.* But note, all this is to be understood in case of slow or equable 

 motion of the body raised; for in quick, accelerated, or retarded motions, the 

 vis inertise of the matter moved will make a variation. 



In comparing the effects produced by water-wheels, with the powers producing 

 them; or, in other words, to know what part of the original power is necessarily 

 lost in the application, we must previously know how much of the power is spent 

 in overcoming the friction of the machinery, and the resistance of the air; also 

 what is the real velocity of the water at the instant that it strikes the wheel; and 

 the real quantity of water expended in a given time. From the velocity of the 

 water, at the instant that it strikes the wheel, given; the height of head produc- 

 tive of such velocity can be deduced, from acknowledged and experimented prin- 

 ciples of hydrostatics; so that by multiplying the quantity, or weight of water, 

 really expended in a given time, by the height of head so obtained; which must 

 be considered as the height from which that weight of water had descended in 

 that given time ; we shall have a product, equal to the original power of the 

 water; and clear of all uncertainty, that would arise from the friction of the 

 water, in passing small apertures; and from all doubts, arising from the different 

 measure of spouting waters, assigned by different authors. On the other hand, 

 the sum of the weights raised by the action of this water, and of the weight 

 required to overcome the friction and resistance of the machine, multiplied by 

 the height to which the weight can be raised in the time given, the product will 

 be equal to the effect of that power ; and the proportion of the two products will 

 be the proportion of the power to the effect ; so that by loading the wheel with 

 different weights successively, we shall be able to determine at what particular 

 load, and velocity of the wheel, the effect is a maximum. 



The manner of finding the real velocity of the water, at the instant of its 

 striking the wheel; the manner of finding the value of the friction, resistance, 

 &c. in any given case; and the manner of finding the real expence of water, so 

 far as concerns the following experiments, without having recourse to theory; 

 being matters on which the following determinations depend, it will be necessary 

 to explain them. 



To determine the Velocity of the fVater striking the fVheel, 



It has already been mentioned, in the references to the figures, that weights 

 are raised by a cord winding round a cylindrical part of the axis. First, then^ 

 let the wheel be put in motion by the water, but without any weights in the 

 scale; and let the number of turns in a minute be 60; now it is evident, that 

 were the wheel free from friction and resistance ; that 6o times the circumference 

 of the wheel would be the space through which the water would have moved in 



