166 OF THE POWER OF STEAM, [ SECT . , v . 



to its load in the descent : and the descending stroke should not be slower than 

 gives the maximum of useful effect, because in both cases a considerable loss of the 

 power of the steam is taking place during its action. 



340. In the ascent of the piston it must be evident that it should never 

 acquire a greater velocity than one which the steam can follow, so as to press it 

 with a force nearly equal to the pressure of the atmosphere ; and when the 

 apertures for the steam are arranged for the descending stroke, the ascending one 

 will be regulated by the passage of the steam through the same apertures ; and if 

 this be the case, and in the present construction of these engines it always is so, 

 our inquiry may be confined to the descending stroke. 



341. The descent of the piston, if the effect of the steam alone were con- 

 sidered, it . is obvious, should be determined by the condition which gives the 

 greatest effect by a given quantity of steam ; but it is dependent on the resistance 

 of the water increasing as the square of the velocity, and the decreasing effect of 

 the steam in the simple ratio of the increase of the velocity. 



342. Now we may be allowed to consider the motion an uniform one, as it is 

 nearly so during the greater part of the stroke ; and then when the steam acts at 

 full pressure during the whole of the descent, the velocity in feet per minute should 

 be 98 times the square root of the length of the stroke. 1 



1 Suppose the arms of the beam to be of equal length ; the steam apertures being the same, the 

 ascending and descending strokes should be made in equal times. The greatest possible velocity 

 V will be generated when the resistance to the water in the pumps is equal to the counter weight ; 

 and as the forces in both directions are to be equal, if m be the force producing the motion ; 

 m = the resistance at the velocity V ; and the resistance to motion in pipes being as the square 



of the velocity, ^^- 2 = the resistance at v, to which the counter weight must be equal ; con- 

 sequently, 



m v 



"* 





- 



= the effective power which is to be the greatest possible. 



The maximum takes place when V 2 dv 3 v e dv = o, that is, when 3t> s = V 2 ,ort; = -577 V. 

 To determine V, it may be remarked, that the motion commences with an excess of power, whicli 

 diminishes by the increase of resistance, till the motion becomes uniform ; that the areas of the 

 passages of the valves are only half the area of the pump, and that the mass of matter moved is 

 twice the excess of moving force ; hence, V 2 = 8 s, and since 3 v* = V 2 , it is 



v = v L = 1-633 V s, the velocity in feet per second. 

 3 



When the velocity is in feet per minute, it is 



60 x 1-633 -J~g = 98 >/l. 



