192 Prof. R. Clausius on the Theorem of the Mean Ergal, 



employing for distinction the indices 1, 2, and 3. The loga- 

 rithm which arises from the addition of the three logarithms we 

 will represent by a simplified symbol, putting 



iogU- 2 iog (te)3 , 

 or 



VL=\/M£k (95) 



{4tm 



The quantity 11 is, according to the above, approximately equal 

 to the product (c 1 + c , ,)(c 2 + c , 2 ) (cg + c'g), or equal to the space- 

 content of the vessel ; and since the latter, according to (79) 

 and (81), only differs by the small quantity Nf 7rp 3 or e from 

 the volume V of the quantity of gas under consideration, which 

 is represented by the material points present in the vessel, we 

 may also say U differs but little from V. 



The purpose of this approximate determination of U is merely 

 to give a convenient representation of the signification of the 

 subsequent equations. The exact determination of this quantity 

 can likewise be effected, if beside equation (95) we take into con- 

 sideration equations (91) and (86) and apply them to the before- 

 discussed motion of the material points in the rectangularly 

 parallelepipedal vessel of the dimensions determined by equa- 

 tions (81). 



In consequence of equation (95) we can now put 



log (limits) = 2 log U + 3 log (4m) . 



The last logarithm on the right-hand side is constant; so that 

 its variation =0, and hence we have 



8 log (Ujltallg) = 28 log U, 



by which equation (94) is transformed into 



8U=£T81ogU + S^8c (96) 



To this equation we can at once join two others, which deter- 

 mine the quantities E and U — T. For this we need only, in 

 one case, to add, and in the other to subtract, the variation hT 

 on both sides. We will at the same time suppose, on the right- 

 hand side, 



ST=TSlogT=TSlog^-, 



rim 



and then, to simplify, introduce the letters (£ and Q with the 

 meanings 



