On the Expansion of Elastic Fluids. 343 
same ratio. This is the only modification which a fractional 
grade produces in those that precede it. In all other respects the 
mode of expansion, up to the fractional grade, corresponds to the 
view presented in the foregoing synopsis. 
We are now prepared to construct a formula for the final ve- 
locity of a fluid which expands from any density D to any other 
nsity d. 
Let V be the final velocity; v the velocity due to a height 
equal to one fourth the subtangent of the fluid; » the number of 
zed ae sige 
complete terms of the series 22’ ®’ 16’ &c., which intervene 
between D and d. 
Then vn is obviously the velocity of the grade which precedes 
the fractional grade, if there be a fractional grade. When the 
first grade is fractional we found its velocity to be v/2° Td} 
and we also found that to suit this expression to the case of a 
fractional grade occurring elsewhere in the range of the series, 
we are to substitute for D that term of the series which is 
next greater than d. Now the value of that term will be ane 
Making the substitution accordingly, the expression for the addi- 
tional velocity due to expansion into the fractional grade becomes, 
- 2nd ° ‘ 
after reducing, 7/2 - ST By adding this quantity to vm we 
obtain the final velocity of the fluid, resulting from its expansion 
from any density D to any other density d. Hence the formula is 
PS ay ee 
V=v.nt+V/2: “WanDg . 
When there is no complete term of the series between D and d, 
n=0 and the above formula becomes V = 71/2 ° Da 
When there is no fractional grade, that is, when d is equal to 
some term of the series, that part of the formula beyond n equals 
0, and then the above formula becomes Veen. | ‘ 
From the general principles here developed it is obvious that, 
as in expansion, so likewise in condensation, the transition of an 
elastic fluid from one density to another is not by gradations 
which may be represented by a curve, but abrupt, mstantaneous, 
per saltum vel saltus. Pulses, therefore, which are propagated 
in elastic fluids partake of the same character ; that is, the con- 
densation and subsequent reéxpansion of the successive elements 
through which the wave moves is instantaneous. This fact was 
not known when the article on the propagation of pulses, referred 
to at the commencement of this article, was written. It however 
does not affect the validity of the reasoning in that article. 
