200 WAVES IN LIQUIDS. [CHAP. VII. 



If a = 0, we get the case treated by Thomson, viz. the case in 

 which a mass of liquid oscillates about the spherical form under 

 the mutual gravitation of its parts. The formula (50) becomes 



2 _47r 2 6 2n + l 

 Tn &quot; g 



&quot; It is worthy of remark that the period of vibration thus calcu 

 lated is the same for the same density of liquid, whatever be the 

 dimensions of the globe. 



&quot; For the case of n = 2, or an ellipsoidal deformation, the length 

 of the isochronous simple pendulum becomes J6, or one and a 

 quarter times the earth s radius, for a homogeneous liquid globe of 

 the same mass and diameter as the earth ; and therefore for this 

 case, or for any homogeneous liquid globe of about 5J times the 

 density of water, the half-period is 47 m 12 s *.&quot; 



&quot;A steel globe of the same dimensions, without mutual gravi 

 tation of its parts, could scarcely oscillate so rapidly, since the 

 velocity of plane waves of distortion in steel is only about 10,140 

 feet per second, at which rate a space equal to the earth s diameter 

 would not be travelled in less than l h 8 m 40 8 f.&quot; 



If on the other hand the depth h of the ocean be small compared 

 with the radius of the earth, we have, writing in (50) b = a + h, and 



neglecting squares, &c. of - , 



CL 

 ^ 



a result due to Laplace J. 



For large values of n the distance from crest to crest in the 

 surface represented by (47) is small compared with the radius of 

 the earth. The propagation of the disturbance then takes place 

 according to the laws investigated in Art. 147. The formula (53) 

 then becomes, approximately, 



T 2jra -f- njgh, 



a result which the student who is familiar with the properties of 

 spherical harmonics will easily see to be consistent with (7). 



* Phil. Trans. 1863, p. 610. 

 + Phil. Tram. 1863, p. 573. 

 I Mecanique Celeste, Livre 4 me , Art. 1. 



