﻿Deduction of Thermodynamical Relations. 753 



from the equation 



dU=F<fafL<?0, (1) 



where U is the energy, x the length, F the stretching force, 

 the angle, and L the moment of the twisting couple. The 

 complete equation, however, is 



dJJ=Tds + ¥dx + Lde, (2) 



(s = entropy), from which (1) can only be obtained if ^5 = 0, 

 i. e. if the processes are supposed to be adiabatic. The 

 author's intention is that the changes shall be isothermal, in 

 which case (1) is incorrect. We obtain the correct expression 

 by writing (2) in the form 



d(]J-Ts) ^-sdT + fdx + Jjde, 



and putting dT = Q, when it reduces to 



d(\J-Ts) = Fdx + Ld0. 



As this equation is of the same form as (1), and as the further 

 proof depends on the first side being a complete differential, 

 the author's conclusions are all valid. 



The same error in a different form is contained in the 

 author's reasoning in connexion with the graphical proof of 

 the relation in question : this method of proof was also used 

 by the author in a previous paper * on magnetostriction, where 

 a similar relation is deduced, and here the fallacy appears 

 very openly. It is there explicitly stated, on page 82, that 

 relations of the kind considered can be inferred from the 

 principle of energy (first law) without using the second law 

 at all. But the proof depends on the theorem that in an 

 isothermal cycle no account need be taken of the heat-effects 

 which cannot possibly be inferred from the first law : it can 

 only teach us that the total heat-effect is equivalent to the 

 total work, not that either of them disappears. This follows 



— r =0, which for an isothermal cycle 



simplifies to J c£Q = 0; and this is the theorem in question. 



In connexion with Dr. Housioun's papers I wish to draw 

 attention to a method of deducing thermodynamical relations 

 which in these and many other cases affords considerable 

 simplification. The usual method is starting from the 

 entropy equation in its immediate form to deduce by the 

 properties of complete differentials a relation between dif- 

 ferential coefficients, and then by changing the variables put 



* R. A ? Houstomi, Phil. Mag. [0] xxi. p. 78 (1011). 

 rhil Mag. S. G. Vol. 23, No. 137. May 1912. 3 1) 



