B. Galitzine on Radiant Energy, 117 



whereas, according to Boltzmann, 



dE 



-mi 



T dT-B]. . . . (6a) 



Second Proof. 



Start with B at a distance k from A, and let A have tem- 

 perature T. Let us take T and h as independent variables. 



Consider, now, how much heat dQ must be given to the 

 system when T increases by an amount dT, and h by dh. The 

 work done in this case is Fdh, and 



dQ=d(he) + ~Pdh; 



dH 



dQ=(e+F)dh + h^dT. 



The increase of entropy e?S is therefore 



It follows from this, since according to the second law rfS 

 must be a perfect differential, and since e is a function of T 

 only, that 



dT T~T' " ' K) 



This equation is an immediate consequence of equation (4), 

 from which it may be obtained by differentiation. 

 On integrating equation (7) we obtain 



P=t[c+ f T ^aTJ, 



P=T[c 1+ r f ^rfT]-«. . . . (8) 



To make this formula agree with (5) we must put the con- 

 stant Ci equal to zero, which appears perfectly legitimate. 

 We shall indeed see later that P is proportional to e. If, then, 

 for infinitely small values of T, e is proportional to any power 

 of T, say e = AT n ; the assumption G 1 ^=0 is clearly equivalent 

 to the condition n > 1. 



Third Proof. — This proof jests upon the consideration of a 

 complicated cyclical process, which is the same in principle 

 as that of Boltzmann. I have merely introduced a slight 

 alteration, and drawn further conclusions from the equation 

 which expresses the first law of thermodynamics. 



Suppose the piston B to be at A. Move B through a 

 distance h^ the temperature of A being always kept at T x . 



or 



