i ae q 3 
tangent ; 
On the Expansion of Elastic Fluids. 339 
H-h H “ 
3. Now the velocity due to q 38 to that due to a a as 
H H-h ig 4 
VY 7 to / %} and the times in which the point of ex- 
pansion would move over s with these velocities, are inversely as 
1 
1 
a H-h D 
these velocities, or as Mie to 3 ages But D: > ::H:H—-A, 
1 1 
D D 
and therefore these times are as if q to g,' oF as 2 to 
22. 'The velocity which the force D can impart in these times 
1s as the times respectively. And since it has been shown that 
in the former of these times the velocity 2s will be imparted by 
2x. 
: 2s : 
the force D, we have 2: ./2r::2s: ¥ = 8/22. That is, 
the velocity which the force D is competent to impart to the mass 
Ds in the time in which the point of expansion recedes through 
$, 18 s\/2r. Consequently the momentum which the force D can 
impart in the same time is Ds?./2z. But we have before found 
this momentum to be Ds2z. Therefore Ds?z=Ds? 4/22; wheuce 
t=/2r and r=2. 
Having thus found the absolute value of z, if we substitute 
this value for x in the series (A) we shall have, for the densities 
of the several parts or grades into which the first element will 
have been distributed at the end of the first instant, the respective 
: Bee ot. Dy D 
terms of the following series, viz., 34 8 ig'es & 
We found the velocity of the point of expansion to be that 
which a body will acquire by falling through —g—j the value 
of h being dependent on the value of z. But when 7=2, 
h 
ae. Therefore the absolute velocity of the point of ex- 
: asimaaea” | 
‘ H 
pansion is that which a body will acquire by falling through vu 
or one fourth of the subtangent of the fluid. ; : 
Since the extent of the element is doubled by the first expan- 
sion, the velocity of the first grade will be equal to the velocity 
of the point of expansion, or that due to one fourth of the snb- 
aad an sed additional velocity is imparted in each 
; expansi If then v represent the velocity due to 
