from a Vibrating Body to a surrounding Gas. 407 



and suppose U expanded in a series of Laplace's Functions, 



U +U, + U 2 + ...; 



then substituting and equating the functions of the same order 

 on the two sides of the equation, we have 



and therefore 



^ = _f,^-^)2_^_ /„(,). . . . (n) 



This expression contains the solution of the problem, and it re- 

 mains only to discuss it. 



At a great distance from the sphere the function /„(/*) becomes 

 ultimately equal to 1, and we have 



It appears from (3) that the component of the velocity along the 

 radius vector is of the order r _1 , and that in any direction per- 

 pendicular to the radius vector of the order r~' 2 , so that the lateral 

 motion may be disregarded except in the neighbourhood of the 

 sphere. 



In order to examine the influence of the lateral motion in the 

 neighbourhood of the sphere, let us compare the actual disturb- 

 ance at a great distance with what it would have been if all lateral 

 motion had been prevented, suppose by infinitely thin conical 

 partitions dividing the fluid into elementary canals, each bounded 

 by a conical surface having its vertex at the centre. 



On this supposition the motion in any canal would evidently 

 be the same as it would be in all directions if the sphere vibrated 

 by contraction and expansion of the surface the same all round 

 and such that the normal velocity of the surface was the same as 

 it is at the particular point in which the canal in question abuts 

 on the surface. Now if U were constant the expansion of U 

 would be reduced to its first term U , and seeing that f Q (r) = 1 

 we should have from (11) 



ft TT 



f ~ '• F (c) 



This expression will apply to any particular canal if we take U 

 to denote the normal velocity at the sphere's surface for that par- 

 ticular canal ; and therefore to obtain an expression applicable at 

 once to all the canals we have merely to write U for U . To 

 facilitate a comparison with (11) and (1.2) I shall, however, write 



