OF GASES AND VAPOURS. 193 
perature ¢ of the gas. These three magnitudes are related by 
the known equation (Mariotte’s and Gay-Lussac’s law), 
pr keira ty Yok Ms Behve 92.) 
in which & is a separate coefficient for each gas, and a the co- 
efficient of expansion of the gas on the application of heat; ¢ 
may therefore be simply considered as a function of p and e. 
Its complete differential is then 
dq 
dg 
on which account the above dp and d ¢ depend in such a manner 
on one another that the temperature remains constant. The 
equation (2.) however gives for this case 
dp=k (l+at).de =F de, 
and consequently ce (2. p dq ie ae) ah 
g dp de 
Above, the increase of the quantity of heat refers to m kil. 
gas, but in this case g relates to the unit of the weight; if the 
latter relation be retained, then m.dq should be substituted 
above for dg, and the equation (1.) now becomes 
—.dpt+ - de, 
or dq OG. 4 2. 
: eee rs 
The general integral of this partial differential equation is 
g=F—nui*p. ri 
where F denotes any function of P. But since# =k (1+ af), 
we can likewise substitute for this function a function of ¢, which 
from its arbitrary nature may also be expressed by 
(l+e?) 
a 
F,+k 
- NL Po» 
in which p, is intended to express any constant pressure. Con- 
sequently 
Pg gl gtk er ae ae ee a le 
@ Po 
* Denoting the Napierian logarithm of p. 
