On the Expansion of Elastic Fluids. 337 
each grade respectively into the next. Now since fluids expand- 
ing in equal ratios acquire equal velocities, equal velocities are 
acquired in each of these expansions. If then we find that ve- 
locity and by it multiply the sum of the series (C), the product 
will be the sum of the momenta generated in, or imparted to, the 
parts of the first element in the time in which the point of expan- 
sion recedes through s. 
the quantity Ds be expanded from the density D to the den- 
sity 7’ the space it will occupy will be increased in the inverse 
. * D 
ratio of these densities; and therefore e :D::s:sz. Hence s and 
S& are respectively the spaces occupied by the element before and 
after the first expansion. Now the velocity which the mass Ds 
eaves in this expansion, is obviously that which would carry it 
over the difference between these spaces in the time in which 
the expansion takes place; that is, the velocity imparted in the 
first ex pansion is str—s=s.r-1; ‘and the same velocity is im- 
parted in every other expansion. If then we multiply the sum 
of the series (C) by s.z—1, the product will he equal to the sum 
of all the momenta generated in the parts of the element. This 
product is Ds’x. Therefore Ds?z is the entire amount of mo- 
mentum which the force D is competent ? Ds cae in the time 
in mk the point of expansion recedes through s. 
will now proceed to find another wheel for the mo- 
wenitie, which the force D is competent to generate in the same 
time, in order that by SOMPRIDE it with that just found, we 
may ascertain the value of z. 
et H be the height of a column of fluid of the density D, 
d let H-A be 
Weces weight is equal to the elastic foree D; an 
the height of another column of the same density whose weight 
is equal to the elastic force 2 Then D: t 2H: H—h. Let 
n be the space occupied yo the first Bich aat m ne 
7 the density D, and ms that which ii oepupics 
when expanded to the density = Since the spaces occupied 
by the element in these states are inversely as the densities, 
mn ims: as D::H—A:H, and therefore 
, x ms 
ms—~sn:ms':H-—h:H; whence we obtain H= suit 
In the time in which the point of expansion recedes through 
mn, the element Ds receives a velocity which will carry it over 
$n in the same.time. If then mn represent the velocity of the 
Szconp Serres, Vol. IX, No. 27.—May, 1850. 43 
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