28 PROFESSOR HORACE LAMB ON THE PROPAGATION OF 



For any particular phase of the motion, t varies as x, and the expression (10(5) 

 therefore varies inversely as x. This confirms, so far, our previous results (95) and 

 (98). Hence with increasing distance from the origin the disturbance tends to the 

 limiting form represented by (101). 



Before leaving this part of the subject, it is to be remarked that the peculiar 

 protracted character of the minor tremor which we have found to precede and follow 

 the main shock is to some extent special to the two-dimensional form of the question. 

 It is connected with the fact, dwelt upon by the author in a recent paper,* that even 

 in an unlimited medium a solitary cylindrical wave, whether of the irrotational or 

 equivoluminal kind, is not sharply defined in the rear, as it is in front, but is prolonged 

 in the form of a "tail." In the three-dimeiisional problems, to which we are about 

 to proceed, this cause operates in another way. The internal waves are now 

 spherical instead of cylindrical, and so far there is no reason to expect a protraction 

 of a disturbance which in its origin was of finite duration. But at the surface they 

 manifest themselves as annular waves, and accordingly we shall find clear indications 

 of the peculiarity of two-dimensional propagation to which reference has been made. 

 On the whole, however, it appears that the epochs of arrival of irrotational and 

 equivoluminal waves are relatively more clearly marked and isolated than in the two- 

 dimensional cases. 



PART II. 



THREE-DIMENSION A L PKOBLEMS. 

 10. Assuming symmetry about the axis of z, we write 



w = ^/(x' + ?/), u = _ c v , v y_q (107), 



CT CT 



so that q denotes displacement perpendicular to that axis. 



A typical solution of the elastic equations, convenient for our purposes, is derived 

 at once from Art, 3, if we imagine an infinite number of two-dimensional vibration- 

 types of the kind specified by (25) and (30) to be arranged uniformly in all azimuths 

 about the axis of -., and take the mean. In this way we obtain from (33), with the 

 necessary change of notation, 



A | V~ cos w d(a = _ 



,, = (- A - #B) . -i I e'*"" da = - (A + ifK) J 



* Cited on p. 37 post, 



.. (108). 



