148 PROFESSOR W. THOMSON ON THE 
which is the same as (3) of § 21; and a combination of this with = (7) = 2% 
t 
: dM dN _1 dp 
gives taRRO aT aire 
which is identical with (2) of § 20. It appears, then, that the consideration of 
the case of fluid motion here brought forward as analogous to thermo-electric 
currents in non-crystalline linear conductors, is sufficient for establishing the ge- 
neral thermo-dynamical equations of fluids, and consequently the universal rela- 
tions among specific heats, elasticities, and thermal effects of condensation or 
rarefaction, derived from them in Part HI., are all included in the investigation 
at present indicated. Not going into the details of this investigation, because the 
former investigation, which is on the whole more convenient, is fully given in 
Parts I. and IIL. I shall merely point out a special application of it to the case 
of a liquid which has a temperature of maximum density, as for instance water. 
137. In the first place, it is to be remarked, that if the two vertical branches be 
kept at temperatures a little above and below the point of maximum density, no 
current will be produced ; and therefore if T, denote this temperature, the equa- 
tion F = of ; dt gives m(T,)=0. Again, if one of the vertical branches be kept at 
T,, and the other be kept at a temperature either higher or lower, a current will 
T 
set, and always in the same direction. Hence we + dt has the same sign, whether. 
T be greater or less than T,, and consequently m(é) must have contrary signs for 
values of ¢ above and below T,: which, by attending to the signs in the general 
formulze, we see must be such as to express evolution of heat by the actual current 
in the second vertical branch, when its temperature is below, and absorption when 
above, T,. As the current in each case ascends in this vertical branch, we conclude 
that a slight diminution of pressure causes evolution or absorption of heat, in water, 
according as its temperature is below or above that of maximum density; or 
conversely,—That when water is suddenly compressed, it becomes colder if ini- 
tially below, or warmer if initially above, its temperature of maximum density. 
This conclusion from general thermo-dynamic principles was first, so far as I 
know, mentioned along with the description of an experiment to prove the lower- 
ing of the freezing point of water by pressure, communicated to the Royal So- 
ciety in January, 1850.* The quantitative expression for the effect, which was 
given in § 50 of Part IIL, may be derived with ease from the considerations now 
| aa) 
: ‘/ shows that 
dt 
the specific heat of the water must be greater in the upper horizontal branch 
brought forward. The other thermo-dynamic equation — 
* See Proceedings of that date, or Philosophical Magazine, 1850. 
