On the Expansion of Elastic Fluids. 341 
sion, and the boundaries of the several grades, or parts having 
different degrees of density and velocity, into which the original 
mass Bn has been distributed. 
t 24 
A a% n B a b é d 
——_—+ =f : ‘ ‘ + ee. 
3 C 
Between these points respect- | 
“ns Dork Ve ioe 
ively the densities are 3" 2 ABu ie 6a 
The velocities are v .2v.3v. 4v. bv. &e. 
These points move toward C 
with the velocities ~v 0 v “20 By 4v&e. 
and relatively to each other, and 
to the fluid, with the velocities 02D 2. D 
As corollaries from the preceding investigation we may state 
the following propositions. : 
1. No other gradations of density can exist in front of a column » 
of fluid which is expanding towards a vacuum except those 
which are found by successive divisions of the original density 
by 2 : 
2. The change of density in the fluid in passing from one of 
these grades to the next is not gradual, but instantaneous ; so 
that the grades are constantly separated from each other by a 
mere imaginary plane. 
3. No other velocities can exist among the parts of a fluid 
which is expanding toward a vacuum but such as are multiples 
of the velocity which a body will acquire by falling through one 
fourth of the subtangent of the fluid. 
A. The velocity imparted to the particles of an expanding 
fluid is not the result of a continual and gradual acceleration, 
but of successive instantaneous increments equal to that whic 
a body will acquire by falling through one fourth of the subtan- 
gent of the fluid. 
It now remains to consider the mode of expansion when the 
fluid is not free to expand indefinitely, but has its expansion arrest- 
d at some given density d. , 
It is obvious that if d correspond in value to any of the terms 
of the series, the manner of expansion up to that point will be 
the same as if the expansion were continued indefinitely. There 
will therefore be in the expanding flnid, in such case, so many 
grades corresponding to the terms of the series, as there are of 
complete terms intervening between Dandd. But let us inquire 
what takes place when d does not correspond to any of the terms 
the series. First, suppose d to be greater than the first term, 
sn from what has been before shown, the velocity of the point 
