Mechanical Equation to the Motion of Material Points. 325 



one and the same quantity X the mean ergal, the mean vis viva, 

 the time of a revolution, and the energy. It is of course under- 

 stood that we eliminate X from each two of these equations, and 

 can thereby obtain relations between each two of the four quan- 

 tities named. Let us suppose this carried out by combining 

 each of the first three equations with the last ; we thus obtain 

 three equations, which determine the mean ergal, the mean vis 

 viva, and the time of a revolution as functions of the energy. 

 This method of determination is peculiarly convenient for appli- 

 cation, inasmuch as in every motion the energy has a constant 

 value, which can be at once given, if only the velocity of the 

 moving point in any position is known. 



We now require only to find the form of the function /(X). 

 This, of course, depends on the law under which the force ope- 

 rates on the point. The determination of this function is espe- 

 cially easy when the force is an attractive force proceeding from 

 a fixed centre, and is represented by any function of the distance ; 

 and this case we will now consider*. 



The distance of the moving point from the centre of attraction 

 may be denoted by r, and the function which represents the 

 quantity of the force by F'(r). If we then put 



f¥'(r)dr = Y{r), (10) 



F(r) is the ergal; and by introducing this function in the place 

 of U, equation (6) is transformed into 



FM=/M (ii) 



If now for any special case of the motion the form of the func- 

 tion which satisfies this equation can be found, the same form 

 will also hold good universally. Such a case is when the point 

 moves round the centre of attraction in a circular path, r is 

 then constant, and we need not take the mean value of F(r), but 

 can write simply 



FW=/W (12) 



Further, in this case the velocity also is constant ; and hence in 

 equation (4) we can, in the place of the mean value v 2 , put 

 simply v 2 , by which it is transformed into 



\ = i\/v i = iv. 

 When, however, the velocity is constant, the product iv is equal 



* As the motion of a material point under the influence of a central force 

 need not take place in a closed path, I will once more give prominence to 

 the fact that the following formulae are to be referred only to motions which 

 take place in closed paths. For the application of my equation to other 

 motions, further special analyses would be necessary, which would carry us 

 beyond our present limits. 



