ON THE GENERAL THEORY OF THE STEAM ENGINE. 175 



Consequently, by integration, 



M 







Now, the engine being supposed to have acquired her permanent speed, there can be no 

 progressive acceleration ; the same velocity must recur at the period of each revolution. The 

 velocity at E is obtained by putting x = o, a = o, and is, 



( 



Also, the velocity at His obtained by putting x = 2 p, a. = tr (= 3-14159), and is, 



Again, it is to be observed that the preceding investigation equally applies to the returning 

 motion along H D, provided, in that case, the symbol a. represents the angle H o D, and x' ', 

 which is accentuated for distinction, the retrograde distance B C. If therefore, x' being 

 each nought, the velocity at H be, 



when a = TT, x' = 2 p, the recurring velocity at E will be, 



r). ..... (d). 



Thus it appears that the squares of the three velocities ( ^ )' ( ^ )" ( ^ )'" are 



in arithmetical progression, and that this progression would go on indefinitely if the resistance 

 R did not augment so as to destroy the common difference 2 P p R p IT, and cause the 

 same velocity to recur at each period. It hence follows that so long as the smallness of R 

 renders 2 P p R pv, or2P RTT, positive, the engine will be acquiring additional speed ; 

 and that when R becomes such that 2 P R TT = o, the power will just be capable of main- 

 taining unaltered the periodical movement, and the general speed will not admit of any further 

 increase. For permanent speed we must therefore have, 



R = - P ...... (6). 



7T 



This result shows that the general effect of the moving force P acting obliquely on O D at D 



o 

 is the same as if " P, or very nearly -fa P, were always applied perpendicularly at the same 



77 



point ; and this again is the same as if the force P itself were continually applied perpendi- 

 cularly at a distance of -fa ths of the radius p. This last distance is therefore the effective 

 leverage of the crank. 



It is not to be inferred from this that the effect of any portion of the power is lost. The 

 quantity of power developed by a constant force is measured by the force multiplied into the 



