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



regarded as phsenomena of the same nature, and we may call two 

 transformations which can thus mutually replace one another 

 equivalent. We have now to find the law according to which 

 the transformations must be expressed as mathematical magni- 

 tudes, in order that the equivalence of two transformations may 

 be evident from the equality of their values. The mathematical 

 value of a transformation thus determined maybe called its equi- 

 valence-value (Aequivalenzwerth). 



With respect to the direction in which each transformation is 

 to be considered positive, it may be chosen arbitrarily in the 

 one, but it will then be fixed in the other, for it is clear that the 

 transformation which is equivalent to a positive transformation 

 must itself be positive. In future we shall consider the trans- 

 formation from work to heat as positive, and therefore the trans- 

 mission of heat from a higher to a lower temperature will be also 

 positive. 



With respect to the magnitude of the equivalence- value, it is 

 first of all clear that the value of a transformation from work 

 into heat must be proportional to the quantity of heat produced, 

 and besides this it can only depend upon the temperature. 

 Hence the equivalence- value of the transformation of the quan- 

 tity of heat Q, of the temperature t, from work, may be repre- 

 sented generally by 



wherein f{t) is a function of the temperature, which is the same 

 for all cases. ^VTien Q is negative in this formula, it will indi- 

 cate that the quantity of heat Q is transformed, not from work 

 into heat, but from heat into work. In a similar manner the 

 value of the transmission of the quantity of heat Q, from the 

 temperature /, to the temperature t<^, must be proportional to 

 the quantity transmitted, and besides this, can only depend upon 

 the two temperatures. In general, therefore, it may be ex- 

 pressed by Q . F(^^^ Q^ 



wherein F(/i, t^ is a function of both temperatures, which is 

 the same for all cases, and of which we at present only know 

 that, without changing its numerical value, it must change its 

 sign when the two temperatures are interchanged ; so that 



^{t^tx)=-nh.h) (4) 



In order to institute a relation between these two expressions, 

 we have the condition, that in eveiy reversible circular process 

 of the above kind, the two transformations which are involved 

 must be equal in magnitude, but opposite in sign ; so that their 

 algebraical sum must be zero. For instance, in the process for 

 a gas, so fully described above, the quantity of heat Q, at the 



