36 PROFESSOR HORACE LAMB ON THE PROPAGATION OF 



Hence, by (7), we have, at a great distance CT, 



*-- w< ^go). 



. --., . 



2/A V TTKiy 2p. TTK7S 



This may be compared with (77). The vibrations are elliptic, with the same ratio of 

 horizontal and vertical diameters as in the case of two dimensions ; but the ampli- 

 tude diminishes with increasing distance according to the usual law CT"* of annular 

 divergence. 



In the same manner we obtain, in the case of an internal source of the type (142), 



. . . . (161), 



where the factor e~"'-' shows the effect of the depth of the source. 



The expressions for the residual disturbance might be derived from the formulae 

 of Art. 8 by the same artifice. Without attempting to give the complete results, 

 which would be somewhat complicated, it may be sufficient to ascertain their general 

 form, and order of magnitude, when ACT and kvs are large. To take, for example, the 

 parts due to the distortioual waves, if we perform the operation id/dx on the 

 second terms of the unnumbered expressions which occur between equations (89) 

 and (90), above, and then replace x by CT cosh u, the more important part of the 

 result in each case is 



e- ika " x>i * tt /(kvr cosh u) 3 - 2 , 



multiplied by a constant factor. This result is to be substituted for the definite 

 integrals with respect to f which occur in (149); the corresponding terms in 

 i/o and IV Q are therefore of the types 



1 r e- ikacosh "du _ md 1 r x e 

 o 1 Jo cosh M* CT' Jo 



(cosh M)* (CT)' o (cosh w) 3 



respectively. By the method by which the asymptotic expansion (7) of the 

 function D () is obtained, it may be shown, again, that these terms are ultimately 

 comparable with 



where the time-factor has been restored. In the same way, the terms in <? and i0 

 which correspond to the expausional waves are ultimately comparable with 



The attenuation with increasing distance is much more rapid than in the case of the 



