164 OF THE POWER OF STEAM, [SECT. iv. 



It must be evident that a regulated or uniform velocity cannot be greater than 

 half the velocity a falling body would acquire in descending the length of the 

 stroke ; because with any other velocity the mass moved would not be capable of 

 receiving and imparting equal quantities of motion in equal times, a circumstance 

 essential to the uniform motion of an engine moved by an uniform force. 



333. As at the greatest possible velocity an engine has no useful power, and, 

 on the other hand, if the resistance be equal to the pressure of the steam, it will 

 have no velocity, there must be an intermediate velocity, which is the best possible 

 for the engine to work with ; and this velocity is one-half the greatest uniform 

 velocity. 1 Now the velocity a falling body would acquire in descending through 

 the length of the stroke is equal to eight times the square root of the length of 

 the stroke, in feet per second ; therefore the velocity which corresponds to the 

 maximum of useful effect, being one-fourth of this velocity, is twice the square 

 root of the length of the stroke in feet per second, and 120 times the square root 

 of that length in feet per minute. 



334. Hence, for engines regulated by a fly, if the pressure on the piston were 

 the same throughout the stroke, the best velocity for the piston in feet per minute 

 would be 120 times the square root of the length of the stroke in feet. 



That is, if the length of the stroke be 4 feet, the square root of 4 being 2, the 

 velocity for a 4 feet stroke is 2 x 120 = 240 feet per minute ; but the action of the 

 valves not allowing this perfection, almost all engines belong to the next case. 



335. If the steam act expansively, the velocity must be less, because the 

 pressure on the piston varies, and the uniform motion the steam would generate in 

 the length of the stroke would be less. 



336. In a steam engine where the steam acts expansively, the supply of steam 



being cut off at the part of the stroke, the best velocity for the steam piston will 

 be found by multiplying the -- part of the length by ? added to 2'3 times the 



1 Let V be the greatest uniform velocity ; m the force producing it ; and w = the mass of 

 matter by which it is rendered uniform ; v being any other velocity. In this case m v w v = 

 the effective action ; and since m v = V w, we have 



m V v mv* 



for the effective part, which is to be the greatest possible. 



The differential gives V dv 2 v dv = o when the expression is a maximum ; whence, 



