Chase.] o04: fjuiy 21, 



couaterbiihinced by some opposing force or forces, so as to produce relative 

 rest ; or the motion must be maintained at a constant distance from tlie 

 centre, so as to produce circular revolution ; or, if the opposing force re- 

 sults from the transformation of interfering revolutions, the two conditions 

 may be combined so as to produce rotation. The velocity of circular revo- 

 lution = |//r; the velocity of rotation = Vf?' -^ n. The value of n may 

 be found by experiment, or, if the aggregating impulses are homogeneous, 

 it may be easily calculated, as will be subsequently shown (V-VII). 



II. If the pressure is varying, and so exerted as to produce radial mo- 

 tion, directly towards the centre, the velocity acquired, at any given dis- 



tance « may be found by the equation v = < > Therefore, if 



^' i P } 

 the theory of Boscovicli were true, at the centre, where f) =0, Vp would 

 be infinite. 



III. If the varying pressure, or pull, leads to i)erpetual radial oscilla- 

 tions, synchronous with perpetual circular oscillations under constant pres- 

 sure, or pull, the extent of the radial excursions from the centre is 2r, and 



2 ,- 

 the mean velocity of radial oscillation (Postulate 14) is ~ -y//'"- The 



equation ~a//'* = \ ; gives p = 3— = 1.4232r. 



IV. Since the vis viva of a moving particle varies as — , the vis viva at the 



r 



radius of average radial velocity in a rectilinear orbit, : the vis viva at the 



radius of synchronous circular velocity : : 1 : 1.4232. 



V. If the constrained synchronous rotation of particles in a spheroid, 

 and the free revolution of exterior particles, are due to the same primitive 

 sethereal impulses (I), the uniform velocity of those impulses is the 

 limiting velocity, towards which both motions tend when their circu- 

 lar paths are indefinitely diminished. Let, therefore, v^ = v/'* OC 



"X — = velocity of free equatorial revolution ; v.^ = — ~ oc ~ = con- 

 r n r 



strained velocity at the same point =: velocity of superficial equatorial 

 rotation. Tlien «g = «, == u^ = nvi = n-v.^ = n \' fr = limit- 

 ing tangential velocity, both of revolution and of rotation, under a reduction of 

 spheroidal volume to an equatorial radius Under such reduction, all 



TO* 



the particles in the equatorial plane would have the velocity of free revo- 

 lution, or of perfect fluidity. 



♦Since /OC ^,, the substitution of 2r for r(, gives /?• (2r — ,") = 2/j,ro 

 7** 



