342 On the Expansion of Elastic Fluids. 
of expansion is that which a body will acquire by falling through 
a when H -/ is the height of a column whose weight is 
equal to the elastic force of the expanded fluid; also that the 
velocity of the point of expansion is that due to the height 7 
a D re 
when the expansion is from D to 5° These velocities are as 
eo eh e " 
q to \’ —g~ and since H : H—A::D : d, those velocities 
D d D 
are as / q to gh 5° The velocity due to g isv. Hence we 
D d 2d 
have Vi : NA : v\/ Fy =Vvelocity of the point of ex- 
pansion in this case. 
Let us next find the velocity of the fluid. The times of run- 
: D 
ning over s by the point of expansion, with the velocities V7 
* 
q : . . *. 
and g are inversely as these velocities; and the velocities 1m- 
parted tothe mass Ds in these times are as the products of the times 
by the respective forces. When the velocity of the point of ex- 
' D D ae 
pansion was / q the force was 5 and the velocity of the fluid 
wasv. The force in the present case is D-—d. Hence we have 
D 
2 D-d ~ : 
a ge ; Wis 1ViV/2 ° ae = velocity of the fluid in this case. 
4 2 
Secondly, suppose the value of d to fall between any two con- 
secutive terms of the series. It is obvious that we have now only 
to substitute in the expression last found that term of the series 
which is next greater than d for D, and it will then express the 
acceleration due to expansion from the last complete term into the 
fractional grade. 
o find the retrogressive velocity of the point of expansion, 
relatively to the fluid, in the grade which precedes the fractional 
grade, we must make the like substitution of the last complete 
term for D in the quantity Ng found above. The retro- 
gressive velocity of the point of expansion in the grade which 
> fractional grade i er than in the o' 8, 
precedes tl is great ther 3! 
and of course that grade will be shorter than the others 1m the. 
