XIX. On the Motion of a small Sphere acted upon by the Vibrations of 

 an Elastic Medium. By the Rev. James Challis, M.A., Plumian 

 Professor of Astronomy in the University of Cambridge. 



[Read April 26, 1841.] 



It is proposed in this Essay to give a mathematical investigation re- 

 specting the motion of a small solid sphere submitted to the dynamical 

 action of the vibrations of a medium so constituted that the pressure {p) 

 and density (p) are related to each other by the equation p = ofp, a 2 being 

 a certain constant. 



1. For this purpose it will be convenient to obtain, first, the equations 

 which apply to the motion of such a medium when directed to or from a 

 centre, whether the centre be moving or stationary. 



Conceive P to be a fixed point in space at which the motion of the fluid 

 is directed to or from a moving centre C. Describe about C as a centre a 

 spherical surface always passing through the point P, and concentric with 

 this another passing through P', a point in CP produced. Let, at a given 

 time t, CP = r, and CP' = r, or r + $r, $r being supposed very small. 

 Conceive now a conical surface, with an indefinitely small vertical angle, to 

 have its vertex at C, and its axis coinciding with CPP, and let it always 

 include the same portion (*»*) of the interior spherical surface. Then if 

 a = the velocity of the centre C resolved in the direction of r, the radius 

 CP at the time t + t, (t being very small) will become r ± ax, and CP' 

 will become r + Ir ± ar, the interval $r being supposed not to vary with 

 the time. Hence the portion of the exterior surface included by the 



Vol. VII. Part III. Oo 



