452 
PEOFESSOE STOKES ON THE COMMUNICATION OF VIBEATION 
At a great distance from the sphere the function f n (r) becomes ultimately equal to 1, 
and we have 
<s=— v . (12) 
r F„(c) v ' 
It appears from (3) that the component of the velocity along the radius vector is of the 
order r~\ and that in any direction perpendicular 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 disturbance 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 at 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 0 , and seeing 
that f 0 (r ) = 1 we should have from (11) 
c 2 IT 
/R — pim(at-r+c) ^0 
r 6 FoW 
This expression will apply to any particular canal if we take U 0 to denote the normal 
velocity at the sphere’s surface for that particular canal ; and therefore to obtain an 
expression applicable at once to all the canals we have merely to write U for U 0 . To 
facilitate a comparison with (11) and (12) I shall, however, write 2U re for U. We have 
then 
<s = — - . . (13) 
r F 0 (c) ' y J 
It must be remembered that this is merely an expression applicable at once to all the 
canals, the motion in each of which takes place wholly along the radius vector, and 
accordingly the expression is not to be differentiated with respect to Q or a with the view 
of applying the formulae (3). 
On comparing (13) with the expression for the function <p in the actual motion at a 
great distance from the sphere (12), we see that the two are identical with the exception 
that U M is divided by two different constants, namely F 0 (c) in the former case and F n (c) 
in the latter. The same will be true of the leading terms (or those of the order r -I ) in 
the expressions for the condensation and velocity*. Hence if the mode of vibration of 
* Of course it would be true if the complete differential coefficients with respect to r of the right-hand 
members of (12) and (13) were taken, but then the former does not give the velocity v! except as to its leading 
term, since (12) has been deduced from the exact expression (11) by reducing f n (r ) to its first term 1 ; nor again 
is it true, except as to terms of the order r -i , of the actual motion of the unimpeded fluid that the whole velo- 
city is in the direction of the radius vector. 
