18 Prof. R. Clausius on the Relations between the 



we have hitherto written as functions of p, can just as well be 

 regarded as functions of q ; and for this we have only to trans- 

 form the relation between J and J { expressed in (31) so that it 

 shall contain q in the place of/?. 



Equation (31) can be written in the following form : — 



^->*0*i0 (37) 



Differentiation of the logarithms gives 



J 2 dp Ji dp \ J /" 



Here J t can be taken away from both denominators ; and, ac- 

 cording to (36), q can be substituted for pJ. Further, we can 

 transform the two differential coefficients according to p into such 

 according to q by employing the general equation 



dZ _ dli dq 

 dp ~~ dq dp 



wherein the differential coefficient ~, which occurs on both sides, 



dp 3 



can be omitted. We thus obtain 



fr'ii® ■■■■■ <»> 



This is the relation sought between J and J^ It is seen that 

 the new equation, in relation to J 15 -~> an( i Q> nas the same form 



as (37) in relation to log Jj, log-y, andjo. 



8. In the preceding, for central motions in which the attrac- 

 tive force is proportional to any power of the distance, a series 

 of formulae is given which represent the times corresponding to 

 the motion-periods, and various mean values, as functions of two 

 easily determinable quantities. As these formulae are scattered 

 among the equations applied to their derivation, it will be advi- 

 sable, for the sake of a more convenient review of them, to 

 briefly recapitulate them in juxtaposition. 



E denoting the energy of the moving point, and c twice the 

 value of the area described by the radius vector during the unit 

 of time, we first form the following two quantities : — 



™»±i) E i4 



p U(n + 3) J 



* I n + 2> J V k nil? 



