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



When, applying the ordinary laws of gases, the work is deter- 

 mined which is accomplished in a change of state, it is found 

 that the factor J-T is the correct one, and consequently that 

 the earlier equation, while in every thing else its form is correct, 

 does not in this point correspond to the reality. 



I must confess that this divergence of the then existing for- 

 mulae from the reality (to which my attention was first turned 

 subsequently) has occasioned me great difficulties. Through it I 

 was obliged to begin the entire mechanical treatment of the 

 subject de novo, and to carry it out in a more generalized manner. 

 In doing so I convinced myself that my earlier equations were 

 certainly sufficient for motions of points in closed paths, but 

 that new considerations, not yet instituted in mechanics, were 

 requisite for that extension of the equations which would make 

 them suitable to be applied to a system of points that do not 

 move in closed paths. By this investigation I arrived at the 

 theorem of the mean ergal, which, because it appeared to be of 

 general importance in mechanics, I published in a separate me- 

 moir, and which in the present one I have applied to the mole- 

 cular motions of gases; at least I have replaced these motions 

 by others which have the same mean ergal and the same mean 

 vis viva, and with which the variables and time-intervals which 

 satisfy the conditions of the theorem can be readily obtained. I 

 think that this result is peculiarly adapted to show how neces- 

 sary it is to contemplate from this new point of view the second 

 proposition of the mechanical theory of heat, if we wish to 

 reduce it to mechanical principles. 



§ 17. The molecular motions of gases are frequently con- 

 sidered in this way : — The molecules are represented simply as 

 elastic balls which exert no forces on one another except after 

 their surfaces have come into contact, when on a still nearer 

 approach they repel one another with a rapidly increasing force. 

 This representation is defective, even for the case in which, 

 neglecting the relative motions of the constituents of a molecule, 

 we wish to take into consideration only the motions of the centres 

 of gravity of the molecules, because, as we must conclude from 

 certain phenomena, the molecules not merely repel, but at some- 

 what greater distances also attract each other. Still the con- 

 ception may serve to give an idea of the nature of the motion 

 and to explain some characteristic properties of gases. It will 

 therefore not be devoid of interest if we develop a little more 

 completely the equation for the case where there exists between 

 the molecules only a repellent force which first becomes sensible 

 when they arrive at a certain degree of proximity, but then, as they 

 approach still closer, increases very rapidly. 



First replacing, as before, the actual motions of the molecules 



