148 PROFESSOR W. THOMSON ON THE 



which is the same as (3) of § 21; and a combination of this with -j- \—\ = *~ \ 



dU <ZN_l dp 

 2 lves dt do ~J dt' 



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 III., 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 III., 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 rt denote this temperature, the equa- 

 tion F = / — dt gives n(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 



rt n 



set, and always in the same direction. Hencey — dt has the same sign, whether 



"o 



T be greater or less than T , and consequently n (t) must have contrary signs for 

 values of t above and below T : which, by attending to the signs in the general 

 formulae, 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, Ave 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 III., may be derived with ease from the considerations now 



dt 



the specific heat of the water must be greater in the upper horizontal branch 



* See Proceedings of that date, or Philosophical Magazine, 1850. 



d 



brought forward. The other thermo-dynamic equation -I 1 = x l J shows that 



Z CLZ 



