Prof. R. Clausius on the Theorem of the Mean Ergal. 191 



If a girder have loads in several spans, we find the moments 

 due to the load in each span separately, and add the results. It 

 will thus be seen that the formulae (1) to (7), in connexion with 

 the equation of the elastic line, contain in a form easy for use the 

 whole theory of continuous girders over level supports. 



XXIII. On the Theorem of the Mean Ergal, and its Application 

 to the Molecular Motions of Gases. By R. Clausius. 



[Concluded from p. 11 7-] 



§ 15. nPHE quantities |*j, u 2 , tt 3 in equation (94) we will first 



-i- subject to a closer consideration. 

 According to equation (91), 



fee 

 log u.z*f(z)dz. 



Here u has the meaning given in (86), namely 



U = m \dt) 1 ' 



-r- ) is constant for most of the time and only during 

 the brief periods of collision has different values, i \ / ("^\ : e 



V \dt) 1S 



approximately equal to the projection, referred to the ^-direction, 

 of the path travelled by the point during the time i; and as, 

 further, the point during this time runs once forwards and back 

 again between the two walls perpendicular to the ^-direction, 



l \f (—) is approximately equal to 2(c + c'), and consequently 



u approximately =4m(c-fc / ) 2 . This holds for all the points, 

 notwithstanding the inequality of z ; and if we take account of 

 this in the above formula for log u, and recollect that 



f 



z*f(z)dz = l, 



we recognize that log it must be nearly equal to \og4m(c + (/) q . 

 The latter result we will bring into a form more convenient for 



what follows, by saying that \ / — is nearly equal to c + c*. 



V 4m 



This result, found for any coordinate-direction, we can of course 

 express also for the three directions of the coordinates singly, 



