Prof. Challis on Hydrodynamics. 219 



which, it may be remarked, is the same that would be obtained 

 if cf>{t) be supposed to vanish. This equation gives by integra- 

 tion 



. ¥(z — /cat) 

 p — i= . > 



so that 



These general results being applied to the case of central 

 motion for which r = r', and r being substituted for z, we have 

 to the first approximation 



The last equation shows that if V be supposed to consist of parts 



Vj and V 2 such that Y 1 = fcacr and V 2 = -^-~ , these two motions 



may coexist, and that the former is propagated with the velocity 

 ica and varies inversely as the square of the distance, while the 

 latter, not being accompanied by condensation, obeys the law of 

 the central motion of an incompressible fluid. 



The laws of central motion that are independent of arbitrary 

 circumstances having been thus obtained, we may now revert 

 to the equations that apply to given arbitrary disturbances pro- 

 ducing central motion. These equations for propagation from 

 the centre are 



y - f ( r ~ Kat ) f( r ~ Kat ) 

 r r z 



fCr-xat) 

 r 

 The difficulty that previously occurred in the application of these 

 equations may now be met by the obvious inference from the 

 foregoing argument that they are not proper for determining 

 laws of central motion, but apply exclusively to the parts of the 

 fluid which, both as regards time and space, are immediately 

 acted upon by the arbitrary disturbance. For example, if the 

 fluid be continuously impressed with a given velocity mcf)(t) 

 at the distance c from the centre, we shall have 



mm = f'(^) _f(c-™t) > 



which equation, by putting T for f(c—/cat), is convertible into 

 this, 



^ +^T + /cflCTwA(0=O. 



dt c r w 



