218 MILNE ART. F 



Now choose A'y = — Ay, thus restoring the initial volume. Then 



(Ai/^ + ^'^P)l, , < - pAy + \ Vd'v 



J v„ — Av 



< 



where po denotes the initial pressure. 



/dp\ 

 At this point we encounter a difficulty. For I 7" ) is negative, 



and so we have apparently only established that the total incre- 

 ment in \p, namely (A + A')\p, is less than a positive quantity. 

 We have thus apparently not proved that it is negative. But 

 if we examine the argument, we see that the original increment 

 in f , namely A^, must be in general of the order Ay, and in fact 

 there exists a constant c such that Af < clAy|, where c < 0. 

 This means that (7) may be replaced by 



A^ < —pAv + c I Ay I, 



whence 



(A -\r A') ^ < c\Av\ - (jX'h ^^"^'• 



Hence in general 



[(A + A>]^. < 0, 



which contradicts (5) and so establishes our result. The 

 difficulty here encountered demonstrates the great need for 

 care in establishing thermodynamic inequalities. The reader 

 may find it necessary to overcome a similar difficulty in the 

 proof left to him in the preceding section. 



It is less difficult to prove the converse. Suppose now that 

 we are given a state for which 



{AlP)t.r < 0. 



