22 PEOFESSOR HOEACE LAMB ON THE PEOPAGATION OF 



It is to be noticed, in all our formulae, that if we write 



= p0, k = pa, k = pb, K = pc, 



J -- ' . 



the symbol p which determines the frequency will disappear, except in the 

 exponentials; this greatly facilitates the desired generalization by means of 

 FOURIER'S theorem. Thus, in the case of a concentrated vertical pressure Q (t) acting 

 on the surface, the formulae (73) and (74) lead to 



- 6 2 ) 2 - 40 2 v /(0 2 - a 2 ) 



The definite integrals represent aggregates of waves, of the same general type, 

 travelling with slownesses ranging from a to b, and from b to oo , respectively. 



If we suppose that Q (t) vanishes for all but small values of t, it appears from 

 (92) that the horizontal disturbance at a distance x begins (as we should expect) 

 after a time ax, which is the time a wave of expansion would take to travel the 

 distance ; it lasts till a time bx, which is the time distortional waves would take to 

 travel the distazice ; and then, for a while, ceases.* Finally, about the time ex, comes 

 a solitary wave of short duration (the same as that of the primary impulse) represented 

 by the first term of (92). This wave is of unchanging type, whereas the duration 

 of the preliminary disturbance varies directly as x, and its amplitude (as will be seen 

 immediately) varies inversely as x. 



If we put 



()cfe .......... (94), 



the integration extending over the short range for which Q is sensible, the 

 preliminary horizontal disturbance will be given by 







provided 



U (ff) = - 



r--- ......... (95), 



irp.bx \x 



(20 2 - b*Y + 160* (ffi - a 2 ) (W - ffi) 



where a < < b. The following table gives the values of U (0) for a series of values 

 of 8/a, on the hypothesis of X = /A, or b/a = 17321. 



This temporary cessation of the horizontal motion is special to the case of a normal impulse. If 

 the impulse be tangential, the contrast between the horizontal and vertical motions, in this respect, is 

 reversed. 



