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



increases that it is sensible only in proximity to the side ; while 

 it may be neglected at greater distances, and especially in the 

 centre of the parallelepipedal space. 



In order to determine the position of the point, let rec- 

 tangular coordinates x v x 2 , x 3 , be introduced, the origin of 

 which coincides with the centre of the parallelepipedon, and their 

 directions are parallel to its edges. 



If we now consider any one of the coordinates (which may be 

 designated by x without index), and if c denote the distance of 

 the two sides perpendicular to this coordinate's direction from 

 the centre, the distances of the movable point from these two 

 sides will be c—x and c + x. The forces exerted by the two 

 sides upon the point, of which the first acts along the negative 

 and the second along the positive ^-direction, we will represent 



na n , not n 



— m- r—rr and m 



(c—xY l+1 {c + x)^ 1 



m denoting the mass of the material point, n a positive number, 

 and a. a constant very small in comparison with c. We hence 

 obtain, for the total force-component X acting along the ay- 

 direction on the point, the equation 



~ r not 71 not 71 i ,^'. 



X = m [-J^y^+J^y^\- ■ ■ ■ (20) 



This gives, for the part of the ergal that refers to the ^-direc- 

 tion, the expression 



f a" g "I 



L{c— x) n ~*~ {c + x) n \' 



m 



Since, according to (20), the force-component belonging to 

 one coordinate-direction depends only on that coordinate and 

 not on the two others, we can again, as in the former case, con- 

 sider the motion in each of the directions separately. A further 

 simplification results from the fact that, on account of the 

 smallness of the factor ct n with every position of the point, we 

 need take into account only one of the two terms in the brackets 

 on the right-hand side of equation (20), viz. that one of which 

 the denominator is less than c n+1 . If, then, we wish prelimi- 

 narily to limit our consideration to the motion during a period 

 in which x has only positive values, for this portion of the mo- 

 tion we can employ the following equation : 



v d 2 x nu n 



A=m— r^- = — m^ 



dt 2 (c-x) 



n+l 



