272 Prof. 11. Clausius on a new Mechanical Theorem 



hcmce not be universally admissible to replace the variation 



by the symbol 



_ 8(U-T),_ 

 which represents the variation obtained when the mean value 

 U — T is regarded as a quantity independent of the time and is 

 variated. 



In our equation (20), however, the first of the two variations 

 just mentioned does not itself occur, but only its mean value. 

 This becomes constant for large times, as can indeed be seen 

 from this, that an expression which becomes constant for large 

 times stands on the right-hand side of the equation. Conse- 

 quently the previously mentioned distinction, based on the vari- 

 ability of the variation, ceases ; and we can therefore employ the 

 symbol 8(U— * T) for this constant-becoming mean value of the 

 variation. Thereby equation (20) is changed into 



S(U-T) = 2p?8 log i -f S ~ ^ 



or the equation (II.) which was to be demonstrated. 



9. As an example of the application of the equation, we will 

 take a simple special case for closer consideration. 



Given two material points which, according to any law, attract 

 and, at certain distances, repel each other, and under the influ- 

 ence of this force move about one another. 



As the centre of gravity of the system remains fixed, and the 

 motion of both points takes place in one plane, we can deter- 

 mine their position by two variables — their mutual distance r, 

 and the angle 6 which the line that joins them makes with a 

 fixed straight line. If, namely, the masses of the two points be 

 denoted by m and /^, their distances from their common centre 

 of gravity are 



" , m 



and r. 



ffl-f fx m + jj, 



If, further, by be understood specially the angle which the 

 part of the right line r from the centre of gravity to the massm 

 makes with the positive ^-direction of a rectangular-coordinate 

 system taken in the plane of motion, the rectangular coordinates 

 of the two points can be expressed as follows : — 



" cos V, y l = — — r sm r/, 



m-\-fJL " x m + jA 



m a m . a 



oc q = r cos v, y 9 = r sin 6. 



2 ?n + fi ; J2 m + fx 



