414 Prof. R. Clausius on the Second Fundamental 



which is no longer applicable merely to the changes of condition 

 of a single body, or even to such a series of changes of condition 

 as constitute a cyclical process, but which holds good also for any 

 series whatever of changes of condition that occur in a reversible 

 manner, either of one body or of a number of bodies. This is 

 the theorem known as the Second fundamental Theorem of the 

 Mechanical Theory of Heat, the elucidation of which was the 

 object of the foregoing considerations. It runs as follows : — 



In any process, how complicated soever it may be, whereby one 

 or more bodies undergo any reversible alterations whatever, the alge- 

 braic sum of all the transformations which take place must be zero. 



It will be seen at once that a great degree of similarity ob- 

 tains between this theorem and the first fundamental theorem of 

 the mechanical theory of heat. According to the first funda- 

 mental theorem, heat and ergon are related to each other in such 

 wise that, in order to produce ergon an equal quantity of heat 

 must be expended, and in order to produce heat an equal quan- 

 tity of ergon must be expended. Production and expenditure 

 can, however, be included under a single conception by count- 

 ing expenditure as negative production. The foregoing relation 

 may therefore be thus expressed: — In every process the algebraic 

 sum of the heat and ergon produced is zero ; and, in exact corre- 

 spondence with this, the second fundamental theorem asserts 

 that the algebraic sum of the transformations is zero. 



Consequently, if the first theorem is called the Theorem of the 

 Equivalence of Heat and Ergon, the second may be naturally 

 called the Theorem of the Equivalence of Transformations. 



This second theorem can be represented mathematically by an 

 equation of equal simplicity with that which expresses the first 

 theorem ; and these two equations are the two fundamental equa- 

 tions, whence all further equations which the mechanical theory 

 of heat is capable of yielding must be deduced. 



If the equation which expresses the first theorem, even by 

 itself, leads to an extensive series of important conclusions, it 

 will be easily understood that, by combining with it the second 

 equation, the fruitfulness of the theory must be increased still 

 further to a very considerable extent, inasmuch as the second 

 equation is not only capable, like the first, of leading to new 

 conclusions when taken by itself, but from the combination of 

 the two equations additional equations of a different form can be 

 deduced, and these in their turn lead to still further consequences. 



A series of important results has in fact already been arrived 

 at by applying the second fundamental theorem. I may men- 

 tion as examples : — the determination of the volumes of saturated 

 vapours ; the determination of the quantity of vapour which is 

 precipitated when a saturated vapour expands in an envelope 



