THE 

 LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



[FOURTH SERIES.] 



NOVEMBER 1871. 



XL. On the Application of a Mechanical Equation advanced by 

 me to the Motion of a Material Point round a fixed Centre of 

 Attraction, and of two Material Points about each other. By 

 R. Clausius*. 



I HAVE recently, in investigations on the mechanical theory 

 of heat, advanced a new equation relative to stationary mo- 

 tion f, which is connected with the theorem of the minimum 

 effect, but extends to cases to which that theorem does not ap- 

 ply. I will here introduce the equation only in the forms which 

 are suitable for the considerations here intended, referring the 

 reader to my earlier memoir for further transformations. 



Given a material point moving, under the influence of a given 

 force, stationarily in a closed path. Let m be the mass of this 

 point, x, y, z its coordinates referred to a rectangular system, 

 X, Y, Z the components of the force operating on it, v its velocity, 

 and i the time of a revolution. All these quantities, except the 

 first and the last, are variable in the course of the motion ; but 

 each has a certain mean value for the entire revolution. We will 

 distinguish such a value from the variable quantity by putting a 

 horizontal stroke over the sign which represents the latter; so 

 that, for example, x will signify the mean value of x. 



Now let us imagine the original motion replaced by another, 



* Translated from a separate impression, communicated by the Author, 

 from the Nachrichten von der Kbniglichen Gesellschaft der Wissenschaften, 

 1871, No. 8. 



t " Ueber die Zuriickfuhrung des zweiten Hauptsatzes der meehanischen 

 Warmetheorie auf allgemeine mechanische Principien," Sitzungsberichte 

 der Niederrheinischen Gesellschaft fur Natur- und Heilkunde, 18/0, p. 167; 

 Pogg. Ann. vol. cxlii. p. 433; Phil. Mag. S. 4. vol. xlii. p. 161. 



Phil. Mag. S. 4. Vol. 42. No. 281. Nov. 1871. Y 



