338 On the Expansion of Elastic Fluids. 
point of expansion, sn will represent the velocity imparted to the 
fluid by the first expansion. Consequently, the retrogressive ve- 
locity of the point of expansion must be such that in the time in 
which it passes over any space, the force Lhe may give to all 
the fluid in that space the velocity sn. Hence the point of ex- 
pansion will run over A in the time in which the foree D— - 
will give to all the fluid in & the velocity sm. But the force 
D. ; Bh 
D- 7 is equal to the weighf of all the fluid ink. Therefore the 
point of expansion runs over / in the time in which the mass h 
would in falling by its own gravity acquire the velocity sn. The 
time in which a falling body acquires the velocity sm is to that 
in which it would acquire the velocity of the point of expansion, 
or mn as sn to mn; and the spaces over which the point of ex- 
pansion would run in these times are in the same ratio. 'There- 
fore putting S for the space which the point of expansion would 
run over while a falling body would acquire the velocity of the 
point of expansion, we have sn: mn::h:S, or, sn:ms—sn::hi8; 
hx 
whence we obtain S= eh But we have before found 
hxms : 
H= ‘sn Therefore S=H -h; that is, the point of expan- 
sion will run over H —A in the time in which a falling body will 
acquire the same velocity. Consequently the velocity of the 
point of expansion is that which a body will acquire by falling 
through sm 
If the force D act on the mass H during the time that mass 
would fall through H, it would give that mass a velocity which 
would carry it over 2H in the same time, because the force D is 
equal to the weight of the mass. The mean velocity of a body 
falling through H is thag which will be acquired by falling 
through 7- If then the point of expansion moved with the ve- 
locity acquired by a body in falling through 7 in the time of 
passing over H, the force D would be competent to give to all 
the fluid in H a velocity which would carry it over 2H in the 
same time; and consequently in the time of passing over $ it 
would give to the mass Ds a velocity which would in the same 
time carry it over 2s. But the point of expansion as before 
shown, moves with the velocity acquired by falling through 
ss ais 
