162 Prof. Challis on Objections recently made to 



a cylinder of moderate dimensions, for the sake of simplicity it 

 will be neglected, and the density of the fluid will be assumed 

 to be at each instant the same throughout, and to vary only with 

 the time. Accordingly for the acceleration of the piston we have 



lfi~W+n\~ g ' 3 



and since by hypothesis k C2 p Q ^Mg } and j~ = — , it follows 



Kp Q z x 



that 



d% _ Mag _ 



dt 2 ~~ (M +m)z l g \ 



We may now suppose, in order to make the problem more 



general, that the mass m impinges on the mass M of the piston 



with a given velocity V, and that the two masses subsequently 



move on together. Hence their initial common velocity is 



mV 

 -Tj-F . In this expression so much of the momentum of the 



fluid as is generated by the impact at the first instant is left out 

 of account as being indefinitely small. The same is the case 

 with respect to the impact of one solid body on another, if im- 

 pact be regarded as a short and violent pressure ; for the mo- 

 mentum of the impinging body is altered by degrees, beginning 

 with zero, by the reaction of the other, and it is after an interval, 

 however short, that the permanent alteration of its momentum 

 is effected. In the problem before us, the reaction of the fluid 

 is taken account of with sufficient approximation by supposing 

 the density of the air and its pressure on the piston to be at all 

 times inversely proportional to the space which the air occupies. 

 This being admitted, the integration of the above equation gives 



dz\ . m 2 V 2 , 6 , x , 2Mga , z l 



This equation includes the effect of the impact of the momentum 

 mV, and is applicable whether V is positive or negative. The 

 case of V negative might be practically exemplified by suddenly 

 attaching the mass m to M by a string passing over a pulley. In 

 that case z x is always greater than a. 



If M = 0, the formula gives for the square of the velocity of m x 

 V 2 + 2^(«— ^j), which is the same as that for motion in free 

 space, as it ought to be. For since k' 2 p =Mg, if M = 0, we 

 should have jp = 0, and there would be no pressure to impede 

 the descent of m. 



If V — 0, or the mass m be added to M without impact, the 



