194 Prof. R. Clausius on the Theorem of the Mean Eryal, 



five ; and hence we obtain the following result : — The expression 



— 2 —— Sc represents the external work performed in the change 



of volume of the vessel. At the same time, however, it is sup- 

 posed that the walls suffer the same pressure during displace- 

 ment as while they are stationary — or, in other words, that the 

 change of volume takes place in a reversible manner. 



The six terms of the sum 2 -77- can be immediately reduced 

 to three, because the differential coefficients for each pair of op- 

 posite walls, -7- and -77-, are equal, and hence we can put 



Here the sum c + c' is the distance from each other of the two 

 boundary planes of the parallelepiped which are perpendicular 

 to the coordinate-direction we are considering. Applying this 

 to the three directions of coordinates, we get three terms with 

 the variations of the three sides of the parallelepiped as factors. 

 Now, as our parallelepiped has the three sides a, b, b, which are 

 determined by the equations (81), and as those equations con- 

 tain, besides a and b, only V as a variable, the variations of the 

 sides can be represented by expressions in which there occurs 

 only the one variation BY. After inserting these expressions we 

 can contract the three terms into one ; and if the factor of BY 

 then obtained be denoted by — p } the equation 



S^8c=-/>8V (100) 



results, p representing a quantity which may be supposed very 

 approximately equal to the pressure prevailing in the gas. By 

 means of this equation the form of equations (96), (98), and (99) 

 can be again simplified. 



§ 16. The equations obtained in the last section can be trans- 

 formed so that their agreement w r ith those which express the 

 second proposition of the mechanical theory of heat shall come 

 out still more clearly. Equation (96) can be written thus : — 



SU-S^8c=§Taiogtf. 

 Here BU represents the increment of the ergal, and consequently 

 the internal work, and — X —~r the external work. Hence the left- 

 hand member of the equation expresses the total work performed 



