vibrating in a resisting Medium. 4>63 



The two methods are, in fact, so dissimilar in principle, and 

 in their results, that if one is right the other must be wrong. 

 But after the lapse of some years I am not able to discover 

 any error either in the principle or the details of the method 

 I have employed in the Philosophical Magazine, nor of that 

 in the Cambridge Philosophical Transactions. The object of 

 my present communication is to give a third solution, which 

 applies expressly to vibrations of the ball in a compressible 

 fluid. 



It will be proper to begin with proving generally that the 

 same equations apply to the motion of the fluid when directed 

 to or from a moving centre, as when directed to or from a 

 fixed centre. 



Considering, first, the motion of the fluid to be in the di- 

 rection of radii from a fixed centre, conceive two spherical 

 surfaces described about this centre at the distances r and r' 

 differing very little from each other ; and let the interior one 

 pass through the point at which we consider the motion. 

 Conceive also a conical surface, having its vertex at the 

 centre of the spherical surfaces and its vertical angle indefi- 

 nitely small, to intersect with its axis the interior spherical 

 surface at that point. Let m 2 = the small portion of the in- 

 terior spherical surface included by the conical surface ; then 



m 2 7 n 



— I — = the corresponding portion of the outer surface. It 



will be assumed that during a very small time Bt, the velocity 

 and density of the fluid which passes the area m 2 are uniformly 

 v and p ; and, similarly, that the velocity and density of the 

 fluid passing in the same time the corresponding area of the 

 other surface are uniformly \J and p'. Then the quantity of 

 fluid which passes m 2 in the time 8 t is m 2 pvBt; and that 



which passes the other area in the same time, 



p ! v' h t. 



The increment of matter in the included space is, therefore, 



(r n p' v f \ 



— m 2 Bti — ^— — p v J, the velocities v and v f being reckon- 

 ed positive when directed^/™??? the centre. The space itself 

 is ultimately mr (r' — r). Hence the increment of density Bp 



m°~ 8 t jr' 2 p> v' -1 2 p v) 

 mr (?•' — ;•) 



is equal to — 

 Consequently, 



St 



■ p'v-rpv 



