d /dw\ _ 



Energy and Entropy of a Body. 1 1 



d /dw\ dx dx d /dw\ dx dy dw d*x 



dx\dx) d% dr) dy\dx) d% drj dx d^dy 



d /dw\ dx dy d /dw\ dy dy dw d^y 



dx\dy) dr) di; dy\dy) d% drj dy d^drj 



r d /dwX dx dx d /dw\ dx dy dw d 2 x 



d (div\_ dx\dx) dtj drj dy\dx) dr) d% dx dgdrj 



dg\dr)/~ d/dw\ dx dy d /dw\ dy dy dw d' 2 y 



v dx\dy) d% drj dy\dy) d% dr) dy dgdrj 



If the second of these equations is subtracted from the first, 

 roost of the terms on the right-hand side disappear, and there 

 remain only four terms, which may be thus contracted into a 

 product of two binomial expressions — 



d /dw\ d(dw\_ /dx dy dx dy\Vd/dw\ d /dw\l 

 dn\d^)~d^\d^)) ~ V^|'^"^'^|/L%\^/ dx\~dy)\ 



The expression standing on the left-hand side of this equation 

 is Ef,j, and that contained within square brackets on the right- 

 hand side is E^. Hence we finally obtain 



Mt-g-S-tX wi 



Similarly we may also find 



_/dx dy dx dy\ VJ 



If we replace only one of the variables by a new one — if, for 



instance, we retain the variable x while putting the variable r? in 



place of y, we have in the two foregoing equations a?=f, and 



dx dx 



consequently -^ =1, and — =0, whereby they become 



E„=;|E^a n dE'„=gEV . . . (14) 



If, indeed, the original variables are retained but their order 

 of succession altered, the equations in question take the opposite 

 sign, as may be seen at once from a glance at the expressions (4) 

 and (7); that is to say, they become 



E y ,= -E^ and E'^=-E'„. . . . (15) 



§6. 



We now return again to the differential equations (5) and (8) 

 that have been deduced for S and U. 



