86 M. R. Clausius on a modified Form of the second 



of the bodies between which heat passes, and not upon tlie nature 

 of the body effecting the transformation. 



The demonstration of this theorem, however, is based upon 

 too simple a process, in which only two bodies losing or recei- 

 ving heat are employed, and where it is tacitly assumed that one 

 of the two bodies between which the transmission of heat takes 

 place is the source of the heat which is converted into work. 

 But in this manner, by previously assuming a particular tempe- 

 rature for the heat converted into work, the influence which a 

 change of this temperature has upon the relation between the 

 two quantities of heat remains concealed, and therefore the theo- 

 rem in the above form is incomplete. 



It is true this influence may be determined without great 

 difficulty by combining the theorem in the above limited form 

 with the first fundamental theoI'Sm, and thus completing the 

 former by the adding to it the results thus arrived at. But by 

 this indirect method the whole subject would lose much of its 

 clearness and facility of supervision, and on this account it ap- 

 pears to me preferable to deduce the general form of the theo- 

 rem immediately from the same principles which I have already 

 employed in my former memoir, in order to prove the modified 

 theorem of Carnot. 



This principle, upon which the whole of the following deve- 

 , lopment rests, is as follows : — Heat can never pass from a colder 

 to a warmer body without some other change, connected therewith, 

 occurring at the same time. Everything v/e know concerning the 

 interchange of heat between two bodies of diff'erent tempera- 

 tures confirms this, for heat everywhere manifests a tendency to 

 equalize existing difierences of temperature, and therefore to 

 pass in a contrary direction, i. e. from warmer to colder bodies. 

 Without further explanation, therefore, the truth of the principle 

 will be granted. 



For the present we will again use the well-known process first 

 conceived by Carnot and graphically represented by Clapeyron, 

 with this difi'erence, however, that, besides the two bodies be- 

 tween which the transmission of heat takes place, we shall assume 

 a third, at any temperature, which shall furnish the heat con- 

 verted into work. An example being the only thing now re- 

 quired, we shall choose as the changing body one whose changes 

 are governed by the simplest possible laws, e. g. a permanent 

 gas. Let, therefore, a quantity of permanent gas having the tem- 

 perature t and volume v be given. In the adjoining figure we 

 shall suppose the volume represented by the abscissa o h, and the 

 pressure exerted by the gas at this volume, and at the tempera- 

 ture /, by the ordinate h a. This gas we subject, successively, to 

 the following operations : — 



