APPLICABLE TO THE MOTION OF FLUIDS. 391 



17. Again, suppose a, /3 and R to be constant, and 7 to vary with the 

 time. This is to suppose the surface of displacement in successive 

 instants to be that of a sphere of given radius moving in a direction 

 parallel to the axis of x. We shall have 



and therefore 



Hence iV=^.^; 



and V - N( d ^ + -^ + rf *Y - *~ 7 ^ 2 »- s ~ V rf Y 

 and ^ _iV fe + ^ + tf?j ~ 2^ '~di' 2ie -—R—-dI- 



Now the general equation (10), when no impressed force is supposed 

 to act, and when terms involving higher powers of V than the first are 

 omitted, becomes for the case in which the motion is directed to or 

 from a centre, 



: d*V d*V 2a* dV *a*V 

 «r dt r dr r 



the same equation being applicable whether the centre be moving or fixed, 

 as is shewn in Art. 13. This equation is readily transformed into 



d\Vr ,jd\Vr ZVr 



— a 9 



dt 2 \ dr* t* 



the integral of which obtained by Euler (see Peacock's Examples, p. 473) 

 gives, 



^- -±{Ar-at) + F(r + at)\ + I {f (r - at) + F(r + at)}. 



If the arbitrary function F be supposed to vanish, the motion will be 

 propagated from the centre, and for this case 



rr _ f(.r~ at ) A r ~ a t) 

 — * ■ 



r r 



xx 2 



