Prof. P. E. Chase on the Correlations of Central Force. 199 



f n = nf-L = velocity communicated by n equal impulses 

 or in n instants, the distance remaing constant. 

 Then 



a. If the pressure resulting from/ is constant, it must either 

 be exactly counterbalanced by some opposing force or forces, 

 so as to produce relative rest, or the motion must be main- 

 tained at a constant distance from the centre, so as to produce 

 circular revolution, or, if the opposing force results from the 

 transformation of interfering revolutions, the two conditions 

 inav be combined so as to produce rotation. The velocity of 



— . *Jfr 



circular revolution = >//V: the velocity of rotation = — — 



J J n 



The value of n may be found by experiment : or if the aggre- 

 gating medium is homogeneous, it may easily be calculated. 



/3. If the pressure is varying, and so exerted as to produce 

 radial motion directly towards the centre, the velocity acquired 

 at any given distance p may be found by the equation 



r g = \ - — — — — > 2 . At the centre, p = and r f = x . 



7. If the pressure leads to perpetual radial oscillations syn- 

 chronous with perpetual circular oscillations under constant 

 pressure, the range of the radial oscillations from the centre 



is 2/', and the mean velocity of radial oscillation is - >/fr. The 



7T ' 



2 ,j- (fr(2,—p)\i . 2- 



equation — s/fr= \ J —±- •— V "gives p= -3 — 7- = 1*1232 r. 



5. The vis viva of the total force varying as -, the theore- 

 tical vis viva of average radial velocity : vis viva of synchro- 

 nous circular velocity : : 1 : 1*4232. 



e. By the dynamic theory of gases, the average radial velo- 

 city = velocity of constant volume : by the laws of thermo- 

 dynamics, heat varies as vis viva ; and we have already seen 

 (a) that the velocity of circular revolution = velocity of con- 

 stant pressure. We have thus a mathematical verification of 

 the empirical ratio between heat of constant volume and heat of 

 constant pressure. 



£. If the constrained synchronous rotation of particles in a 

 spheroid, and the free revolution of exterior particles under the 

 influence of any central force whatever, are due to the same 

 primitive impulses («), the velocity of those impulses is the 

 limit of velocity towards or from which both motions tend. Let, 

 therefore, v,= V/V = velocity of free revolution, varying as 



\/-j *'//= — •— = constrained velocity at same point, or 

 V / n 



