SECT. 1.] 



THE STEAM ENGINE. 



19 



nished, shall equal the accelerative force ; and that then again the piston may be 

 retarded the rest of the way. But, independent of friction, we can, notwithstand- 

 ing 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 

 were nothing ; whence we shall without difficulty arrive at some conclusion in this 

 matter. Let M N be the working part of a steam engine cylinder of the usual 

 height, equal in diameter to a shorter one m n and the rarefaction in both of them 

 being supposed the same, A Q=a q may represent the excess of the atmosphere's 

 weight above the column of water; RQ=r r , the resistances to the pistons from 

 the remainder of the steam, and A R=a r the effective forces. Make a k : A K : : 



A RQ 



a n : A N, and at all similar positions, the resistance b c of mn, 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 Newton's Princip. prop. 39.) in the descent of bodies, we have jakcr: 



. K C R : : celerity in k : celerity in K. But these areas being evidently as 

 the corresponding parallelograms k q and K Q, and these again as their heights, 

 the celerities generated are in the subduplicate ratio of a A: to A K, as if the resist- 

 ance had been invariable. 



To apply this to steam engines, if T W be a cylinder of equal content with the 

 cylinder M N, the quantity of water delivered by both will, as observed above, 

 be the same at each lift ; but the cylinder T W is no higher than m n, and their 

 rarefactions are supposed 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 m n : : \/E W : VAN; whence, in any given time, the short cylinder 

 T W will perform more than the longer one M N of equal content, and that in 

 the ratio of their diameters ; for as T E x E W=M A 2 x A N ; and E W : AN:: 

 MA 2 : T E 2 J therefore, v/EW : N/A~N : : MA: T E. And he further remarks, 

 the friction is diminished with the slowness of the motion, because the periphery 

 of the piston increases in a less ratio than its area. 



The result of his whole reasoning is in favour of a short cylinder, and it must 

 be allowed to be ingenious ; but the proper question is, What form of the cylinder 



