characteristic Quantities occurring in Central Motions. 13 

 changed into 



8[F(r) -irF'(r)] =rF'(r)8 log i, 



for which, as rF'(r) =mv 2 , we can write 



m — — 



SF(r) = -£6V-fmv 2 Slog«. 



This is the equation which holds for central motions in closed 

 paths, and here appears as a special case of more general 

 equations. 



6. Having put the equations in such a manner that they 

 are valid for any conceivable law of force, we will apply them to 

 a special group of such laws, viz. to those according to which the 

 force is proportional to any power of the distance. At the same 

 time, however, we will exclude the minus first power, because in 

 the integration it leads to logarithms, and consequently demands 

 separate discussions, which would detract from the compendious- 

 ness of the analysis. 



Accordingly, k and n denoting two constants, the latter of 

 which is different from —1, we will put 



P(r)=kr n , (18) 



whence results 



F( r ) == _*-- f -+i. ..... (19) 



w n+ 1 v ' 



These formulas we have to substitute for F(r) and F'(r) in the 

 above equations. Equation (16) may be selected, which by the 

 substitution is changed into 



* ~\ , kSr^^kr^B log i-m (^jYs log-. . (20) 

 2(w + l) & \dtJ °«j v ' 



For the sake of a more convenient expression, the quantity p 

 shall now be introduced, with the signification 



pn+i—^+I ^21) 



This quantity can be immediately determined when the energy 

 (sum of ergal and vis viva) is known, consequently when we know, 

 for any position of the movable point, its velocity. That is to 

 say, since the mean vis viva is equal to the virial, we have gene- 

 rally, if E denotes the energy, 



E=l>)+i^l>), 



and for our special law of force : — 



* ^ ,. rc + 3 



n + L 2 2(n + l) ' 



