336 On the Expansion of Elastic Fluids. 
1 . ; 
by 5, Then the density of the posterior part of the expanded 
D 
element at the end of the first instant is > ° Now for reasons 
which will soon be apparent, all the other parts of the expanded 
element, whatever may be their present state of density, may be 
considered as having passed first into the density 7° But at the 
D 
same time that the grade =a began to form in front of the column, 
that grade itself must have begun to expand again in the same 
ratio, forming another grade is And at the same time that the 
D 
‘ grade ~ began to form, that likewise must have begun to expand 
in the same ratio forming a grade A and so on ad infinitum. 
The grades therefore will correspond to the terms of an infinite 
series in decreasing geometrical progression. All of them origin- 
ate simultaneously in the first element; and yet every grade 
respectively may be considered as having passed into and out of ¥ 
all the grades which precede it ; inasmuch as each in its origin Is 
aconstituent part of that which precedes it. The fluid which 
all of it pass into the next in the same time; for equal quantities 
by measure expand in equal ratios in equal times; and since @ 
distributed into portions or grades, having their respective den- 
sities corresponding to the terms of the infinite series 
DD: D:*D. DP 
Zghath pee 
If we extend this series backward one term we obtain the series 
DDDD 
D, my zr Pry Pri, &e. (B) 
_ Since equal quantities by measure pass out of each of these states 
in a given time, if s be the space occupied by the original element, 
and if we multiply each of the terms of the series (B) by s, then 
: Ds Ds Ds Ds 
the terms of the resulting series Ds, - aa? De &e. (C). 
will severally express the quantities of fluid that expand from 
