348 PROFESSOR CHALLIS, ON THE MOTION OF A SMALL SPHERE 



velocity of the fluid being supposed to remain the same as before, the 

 effect of change of density will be taken into account by merely sub- 

 stituting p^ for 5 in the equation above. By this substitution let f 

 become f. Then 



f = ^le^ t) + ^{e^ fe-^[t)dt-ke^}. 



Again, the sphere will be acted upon by an additional accelerative force 



arising from the circumstance that the density, and consequently the 



pressure, varies from one point to another of its surface at a given 



time, on account of the variation of density of the fluid in vibration 



with the distance x from the origin at a given time. The pressure 



at all points of a plane perpendicular to the direction of x will evidently 



be the same. Hence, if c?f(x) represent the pressure at any distance x, 



corresponding to the position of the centre of the sphere, it may readily 



d f(x) 

 be shewn that the accelerative force in question is — a 2 3— H^-^, terms 



involving r 2 being omitted. Now 



f(x) = e a ' , and — ^M = : • « "' * <t> [t . 



* ax a 2 \ a) 



Hence, calling this force f", we have, 



As the sphere is solicited by no other forces than those just considered, 



d 2 x 

 f +/" = -Tj ; and consequently, 



at 



Now fe' x'tydt = «' '- { x '(t) - - X "(t)}> nearly: 



U fJh 



and e a = e" v a/ a< " = e ffl «-x I — 1 , nearly. Hence 



