174 Prof. Thomson on the Dynamical Theory of Heat. 

 Heuce equation (2) of § 20 becomes 



dL 



at 1 dp y„„. 



7-\ ^Jdi ^^^^' 



Combiuiug tliis wdtli the conclusion (32) derived from the 

 second fundamental proposition, we obtain 



§+-"=¥ («^)- 



The former of these equations agrees precisely with one which 

 was first given by Clausius, and the preceding investigation 

 is substantially the same as the investigation by which he arrived 

 at it. The second differs from another given by Clausius only in 

 not implying any hypothesis as to the form of Carnot's function/t. 



57. If we suppose /u. and L to be known for any temperature, 



equation (32) enables us to determine the value oi -~ for that 

 temperature ; and thence deducing a value of dt, we have 



dt=t^dp ....... (35); 



which shows the effect of pressure in altering the " boiling- 

 point " if the mixed medium be a liquid and its vapour, or the 

 melting-point if it be a solid in contact with the same substance 

 in the liquid state. This agrees with the conclusion arrived at 

 by my elder brother in his Theoretical Investigation of the Effect 

 of Pi'essure in Lowering the Freezing-Point of Water*. His 

 result, obtained by taking as the value for jm that derived from 

 Table I. of my fonner paper for the temperatm'e 0°, is that the 

 freezmg-point is lowered by -0075° Cent, by an additional atmo- 

 sphere of pressm'e. Clausius, with the other data the same, 

 obtains "00733° as the lowering of temperatm-e by the same ad- 

 ditional pressure, which differs from my brother's result only 

 from having been calcidated from a formula which implies the 



E 

 hypothetical expression J z — ^ for /i. It was by applying 



equation (33) to determine —r- for the same case that Clausius 



arrived at the curious result regarding the latent heat of water 

 under pressure mentioned above (§ 45). 



58. Lastly, it may be remarked that every quantity which ap- 



* Transactions, vol. xvi. part 5. His paper was republished, with some 

 slight modifications, in the Cambridge and Dublin Mathematical Journal, 

 new series, vol. v. — Nov. 1850. 



