characteristic Quantities occurring in Central Motions. 19 



We next introduce two functions of q, denoted by J and J lf 

 which are connected by the following equation : — 



dq ~ q dq\J )' 



Then for the undermentioned quantities the equations written 

 after them are valid : — 

 (1) The mean ergal : 



AT — r-, K , . 



r n+l = n M +' 



« i 1 r 



w + 1 ra + 



(2) The mean vis viva : 



m "2 * n+l 



2 2 



(3) The mean vis viua of the rotatory motion (according to 



equation (24) , replacing p therein by j) : 



™#\ _* »+l£, 



2 r*~2 P J 

 (4) The mean vis viva of the radial oscillatory motion 



f(£K'«(H> 



(5) The period of a rotation : 



(6) The period of -a vibration : 



Besides the above quantities, others can easily be expressed 

 after the above developments — for example, the mean ergal of 

 the rotatory motion and of the vibratory motion ; meanwhile the 

 preceding expressions may suffice. 



9. In all these expressions, only one undetermined function 

 of q occurs, since the two denoted by J and J l can, through the 

 relation subsisting between them, be reduced to one. To deter- 

 mine this function also was not originally included in the plan 

 of my investigation, as I only washed to ' deduce those conse- 

 quences w r hich result immediately from my new mechanical 

 equations ; but after I had produced the above formulse it seemed 

 to me advisable, for the sake of completeness, to undertake at 



C2 



