Intelligence and Miscellaneous Articles. 399 



This first corollary also results immediately from the statement, 

 so luminous in its brevity, of the mechanical theory of heat, given 

 by M. Resal in his Mecanique generate. 



"We can therefore write 



cW=A[T(t>'(T)dT + 'Rdvl 



R being a function of v only, and (p'(T) any function of the tempe- 

 rature. 



Observe now that, in virtue of the second general proposition of 

 the mechanical theory, we have, if T is the absolute temperature 

 and fj. denotes what M. Clausius calls the entropy, 



dQ, = Tdu=T(^.dT+ c ^dv\ 

 \dT dv J 



Introducing these values of dTJ and dQ, into equation (1), we get 



T [^~ A ^' (T) ] Cn: + ( T ^ - AE - A P)^= ' 

 which requires that we have separately 



S-A«/(T)=0, 



T^-AK=Ap = 0. 



dv 



From the first we derive, V being an arbitrary function of the va- 

 riable v alone, 



.-AffCTHJ**] 



We thus get this second proposition : — 



Whatever the body considered, the quantity called by M. Clausius 

 the entropy cannot be any function of the volume and the temperature ; 

 in the same way as the internal heat, the entropy can only be the sum 

 of two functions, — the one, of the volume alone; the other, of the tem- 

 perature alone. 



Consequently the second of the equations obtained gives 



(p + R)V=T, (A) 



which establishes the law above enunciated. Such is the necessary 

 form which connects the pressure, the volume, and the temperature 

 of any body whatever, R and V being two functions of v only. 



We said, at the commencement, that theory had not hitherto 

 supplied any certain and general indication like that which is in 

 question here. On this subject we must make one remark. M. 

 Hirn, in the last edition of his Exposition de la Theorie mecanique 

 de la chaleur, very judiciously divides that theory into two branches : 

 in the first he develops the rigorous consequences of the two funda- 

 mental propositions; in the second he states a great number of 



