160 
ME. J. H. JEANS ON THE VIBEATION3 AND 
interest turns upon the compressibility. The solution which follows is, in its main 
points, very similar to that of Professor Lamb, so that I have not thought it 
necessary to give the steps of the argument in great detail. 
From the symmetry of the solid it follows that the elastic constants \, fi, and the 
density p, will be functions of the single co-ordinate r, the distance from the centre. 
Taking the centre as origin, we shall use rectangular co-ordinates, x, y, z, and shall 
suppose the solid to execute a small vibration, such that the displacement of the 
element initially at x, y, z has components, ^, 77 , The component of displacement 
along the radius will be denoted by u and the cubical dilatation by A, so that 
u = -{^x-\-rjy tz), 
A=+‘is + y. 
dx ^ dy ^ dz 
§ 6 . After displacement the density at x, y, z is 
or, since p is a function of r only, 
dp 
Hence tlie gravitational potential at x, y, z is changed by displacement from Y 
into V ~ E, where E is the potential of the following distribution of matter :— 
(i.) A volume distribution of density 
+ .(A 
(ii.) A surface distriljution of which the surface density is 
(Po - Pi) 
(5), 
this being taken over every surface at which the density changes abruptly, the change 
being from p^ to p^ in crossing the surface in the direction of r increasing. In 
particular this will occur at the outer surface of the solid, the value of p^ in this case 
being zero.* 
§ 7. The potential at rr, y, z after displacement being Y — E, tliat at x -f- ty + rj, 
z t, will be 
.a\ 
0 \ 
0V 
0 .'/ 
dz 
03Y 
07 
+ 
-E-^ 
0E 
A- 
0E _ 0E _ 
d y ^ 0 f “ • • * 
*■ In the investigations on gravitating spheres given in Thomson and Tait’s ‘ Natural Philosophy,’ the 
course of procedure is tantamount to neglecting the volume distribution (4), and regarding E as the 
potential of a surface distrilmtion (5) alone. For this reason the result obtained differs from that of the 
present paper. 
