90 Prof. Clausius on the Internal Work 



another body during the alteration of condition. We have, how- 

 ever, already represented by dQ, the infinitely small quantity of 

 heat imparted to another body by the one which is undergoing 

 modification, hence we must represent in a corresponding man- 

 ner, by — dQ, the heat which it withdraws from another body. 

 We thus obtain the equation 



-dQ = dK + AdL, 

 or 



dQ + dH+AdL=0* (1) 



In order now to be able to introduce the disgregation also into 

 the formulae, we must first settle how we are to determine it as a 

 mathematical quantity. 



* In my previous memoirs I have separated from one another the internal 

 and the external work performed by the heat during the change of condition 

 of the body. If the former be denoted by d\, and the latter by dW, the 

 above equation becomes 



dQ+dK+Adl + AdW=0 (a) 



Since, however, the increase in the quantity of heat actually contained in a 

 body, and the heat consumed by internal work during an alteration of con- 

 dition, are magnitudes of which we commonly do not know the individual 

 values, but only the sum of those values, and which resemble each other in 

 being fully determined as soon as we know the initial and final conditions 

 of the body, without our requiring to know how it has passed from the one 

 to the other, I have thought it advisable to introduce a function which 

 shall represent the sum of these two magnitudes, and which I have denoted 

 by U. Accordingly 



dJJ = dH.+Adl, (b) 



and hence the foregoing equation becomes 



dQ+dU+AdW=0; (c) 



and if we suppose the last equation integrated for any finite alteration of 

 condition, we have 



Q+U+AW=0 {d) 



These are the equations which I have used in my memoirs published in 

 1850 and in 1854, partly in the particular form which they assume for the 

 permanent gases, and partly in the general form in which they are here 

 given, with no other difference than that I there took the positive and ne- 

 gative quantities of heat in the opposite sense to what I have done here, in 

 order to attain greater correspondence with the equation (I.) given in § 1. 

 The function U which I introduced is capable of manifold application in 

 the theory of heat, and, since its introduction, has been the subject of very 

 interesting mathematical developments by W. Thomson and by Kirchhoff 

 (see Philosophical Magazine, S. 4. vol. ix. p. 52^, and Poggendorff's An- 

 nalen, vol. ciii. p. 177). Thomson has called it "the mechanical energy 

 of a body in a given state," and Kirchhoff " Wirkungsfunction." Although 

 I consider my original definition of it (see Pogg. Ann. vol. lxxix. p. 385, 

 and vol. xciii. p. 484), as representing the sum of the heat added to the quan- 

 tity already present and of that expended in internal work, starting from 

 any given initial state, as perfectly exact, I can still have no objection to 

 make against an abbreviated mode of expression. 



