characteristic Quantities occurring in Central Motions. 3 



nates together, but also for each point and each coordinate 



m / dv\ 

 singly. Naming the quantity 9 ( -7- ) the vis viva of the point 



relative to the ^-direction, and in like manner the quantity 



— ^XcT its virial with respect to the #- direction, and considering 

 A 



that the x- may be any direction we choose, we can express the 



proposition in the following terms : — For every point the mean 



vis viva relative to any direction is equal to the virial relative to 



the same direction. 



Now I have been reminded that an equation advanced by 

 Jacobi and another by Lipschitz stand in connexion with the 

 considerations I have instituted. 



Jacobi's equation is found in CrehVs Journal, vol. xvii. p. 121, 

 and in Jacobins Vorlesungen iiber Dynamik, p. 22, where it is given 

 as equation (2). It is there assumed that the forces acting in 

 the system have a force-function, and, still more specially, that 

 this force-function U is a homogeneous function of the kth. di- 

 mension; and the equation relating to it is : — 



in which h is the additive constant belonging to the force-func- 

 tion. It is evident that this equation can only be compared with 

 equation (1 a), and not with (2a) ; and even from (la) it is dis- 

 tinguished, on the one hand, by a very different form (as it does 

 not include the vis viva), and on the other very essentially by 

 holding only for a very limited class of forces, while (la) is valid 

 for all forces. 



Lipschitz's equation is much more general (contained in his 

 memoir " iiber einen algebraischen Typus der Bedingungen eines 

 bewegten Massensysterns"*). He likewise has under considera- 

 tion a system of material points in motion, P p P 2 , . . . P M , which, 

 however, need not be free, but may be subject to conditions 

 which are expressed by equations of a form there more closely 

 defined. He assumes for each point a certain defined position, 

 the coordinates of which are denoted for the point P a by a a , b ffi c x% 

 while P a itself at the time t has the coordinates x a , y u , z a . 

 Further, besides the force-function XJ and the vis viva T of the 

 entire system, he introduces a quantity G, which is determined 

 by the following equation : — 



2 a m x [(x*-a a y + ( yot -b a )* + (z x -ctf]=2G; 



* Journ.fur reine und angew. Math. vol. lxvi. 

 B2 



