22 
MR. J. J. WATERSTON GIST THE PHYSICS OF MEDIA COMPOSED OF 
1 + — that it may sustain the same pressure with the reduced density. But the 
molecular vis viva in the cubical volume is A 3 v 2 : hence we have the proportion 
1 : 1 
: A 3 id : A 3 v 2 ( 1 + ~) ; 
\ 9 n J 
the increment of vis viva in the contained medium is thus A 3 v~ —r. But it has been 
9 n 
S '?2 v * 1 
shown in § 17 that w 2 = ^v 2 , and gn = \v 2 A 3 , hence A 3 v 2 = —. 
Now in § 16 it was shown that to raise the weight of n molecules to the height 
2 iv 2 jgn 2 the vis viva expended was 2 w 2 /n = § v 2 /n ; comparing this, the vis viva 
expended in the act of increasing the volume, with 2 v 2 /n, the increment of vis viva 
required to support the increased volume, it is evident that the ratio is exactly 
one-third. Thus we deduce that to effect one increment of expansion in the volume of 
a medium subsisting under a constant pressure , four-thirds of an increment of vis 
viva are required: one of which thirds is expended in the act and does not appear in 
the medium: the remaining three-thirds, or one increment, appears in the medium and 
assists in sustaining its augmented volume # .XII. 
§ 19. This result is a necessary consequence of the perfect conservation of vis viva 
in the impinging action of perfectly elastic bodies. To enlarge a volume that is pressed 
upon is to raise a weight; is to expend mechanical force ; is to expend molecular vis 
viva, and the last train of reasoning has led us to the relation between the molecular 
force expended and the work performed by it. 
The mechanical value of the whole of the vis viva of the medium may be ascertained 
by the following proportion 
‘6n 
2 o 
?r 
A 3 v- 
A 3 v- 
gn ‘ 
But gn — ^-A 3 v 2 , therefore 
A 3 v 2 
gn 
Thus the vis viva expended in raising n, or the constant pressure, through the in¬ 
crement of the unit of volume, is to that increment as the whole vis viva of the 
medium is to three units of volume. Hence we deduce that if a medium is supposed 
to expand to four times its original volume with its original tension undiminished, it 
will in doing so expend as much as the whole of its original molecular force. This 
may be more concisely expressed by : The molecular vis viva of a medium is equal to 
its tension acting through three times its volume +.XIII. 
* [The corrected version of XII. will be “ To effect one increment of expansion in the volume of a 
medium subsisting under a constant pressure, ^ire-thirds of an increment of vis viva are required; two 
of which thirds is expended in the act, and does not appear in the medium; the remaining three-thirds, 
or one increment, appears in the medium, and assists in sustaining its augmented volume.”—R,] 
t [This is the virial equation applicable under the supposed conditions.—R.] 
