172 Prof. E,. Clausius on the Second Axiom 



elastic walls, from which it rebounds with equal velocity at each 

 impact), we can write simply v 2 for v 2 , and then extract the root, 

 by which the expression i Vv* is changed into iv. This product 

 is equal to the length of the point's path ; and consequently we 

 can say that, in motions with constant velocity, the disgregation 

 (neglecting an additive constant, which in the variation or dif- 

 ferentiation is omitted) is proportional to the logarithm of the 

 length of the path. 



When the velocity is variable, the thing is not quite so simple, 

 because the mean value of the square of the velocity is different 

 from the square of its mean value ; but still it is seen that the 

 disgregation stands in close relation to the logarithm of the 

 length of the path. 



9. Before leaving the motion of a single point in order to pass 

 to more extended investigations, it will be to the purpose to sub- 

 mit to a further special consideration the last of the three causes 

 above mentioned of the conversion of the motion, because we shall 

 thereby have an opportunity of comparing the result of our de- 

 velopment with a well-known and important mechanical theorem. 



We will, namely, assume the conversion to be occasioned by 

 the point having been compelled, instead of the path chosen by 

 itself, to describe another, lying infinitely near it. In this case, for 

 each place in the changed path, compared with the corresponding 

 place in the original path, according to the theorem of the equi- 

 valence of vis viva and mechanical work, the following equation 

 holds good : — 



SU+^S(^)=0. 



__ /YY) 



Accordingly, in equation (19), instead of SU, we can put — -x^v 2 } 

 and hence obtain the following equation: — 



— Bv 2 == -fr- Bv 2 -f- mv 2 B log i, 



from which, by easy transformations, result :— 



mBv 2 + mv* — = 0, 

 i 



2^ + ^ = 0, 



B{v 2 .i)=0 } 



sfW=0 (24) 



This equation is in form the same as that which, for a single 



