188 PHILOSOPHICAL TRANSACTIONS. [aNNO 1751. 



mark too, that if the rarefaction be so complete, that the descent would be 

 greater than the construction admits of, the retardation is augmented by a 

 brachium of the balance pressing on springs. But to say nothing of friction here, 

 we can, notwithstanding this diminution of force by the remainder of steam 

 within the cavity of the cylinder, demonstrate the ratio of the velocities, and the 

 times of descent of the pistons, in cylinders of unequal altitudes, to be exactly 

 the same as if the resistance was nothing ; whence we shall without difficulty 

 arrive at some conclusion in this matter. 



MN is the working part of a steam engine cylinder, of the usual height, equal in 

 diameter to a shorter one m n, fig. 1, pi. 6; and the rarefaction in both of them 

 being supposed the same, AG=aq, RQ=rq, and AR=ar, may represent the ex- 

 cess of the atmosphere's weight above the column of water, the resistance to the 

 pistons from the remainder of steam, and the effective force, respectively, e. g. 

 at the beginning of the descent. Take then every where ak: ak:: an: an, 

 and at all similar positions the resistance b c of m n and force k c on its piston, 

 will be equal to the resistance b c of m n and force k c on its piston ; and by what 

 Sir Isaac Newton has demonstrated (Book 1, Prop. 39,) of the descent of bodies, 

 we have y^akcr: ^akcr:: celerity in k: celerity in k. But these areas 

 being evidently as the corresponding parallelograms kq and kq, and these again 

 as their heights, the celerities generated are in the subduplicate ratio of ak: ak, 

 as if the resistance had been nothing ; and by an obvious enough reasoning from 

 the said proposition, the times also appear to be in the above-mentioned ratio ; 

 which ratio is not any way varied, though the resistance prevails from the inter- 

 secting points o. 



Now, to apply what has been said to the business in hand ; if t w be a cylinder 

 of equal content with the cylinder mn, the quantity of water delivered by both 

 will, as a consequence of the fundamental law of mechanics observed above, be 

 the same at each lift : but the cylinder t w is no higher than n m, and ex 

 hypoth. their rarefactions are equal ; therefore by what has been proved with 

 regard to the times, the time of the piston's descent in t w, will be to that of the 

 piston's descent in mn:: y^Ew: \/an; whence in any given time the broad 

 cylinder t w will perform more than the longer one m n of equal content, and 

 that in the ratio of their diameters; for fe^ X ew = ma^ X an ex hypoth. 

 andEw: an:: ma'^: et% consequently v'ew: v'an:: ma: te. The friction 

 too is diminished with the slowness of the motion, and because the periphery in- 

 creases in a less ratio than does the area of a circle. 



The result of the whole then is in favour of the broad cylinder ; and still the 

 broader the better ; for unless some mechanical considerations should limit the 

 problem, it is evident in a geometrical sense, that there is no limitation. A 

 disadvantage might arise perhaps to the effect of the jet d'eau from thus increas- 



