TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 37 



annular liayleigh waves, so that the latter ultimately predominate.* It is also 

 much more rapid than in the case of elastic waves diverging from a centre in an 

 unlimited medium, where the amplitude varies inversely as the distance. 



14. The generalization of the preceding results, so as to apply to an arbitrary 

 time-variation of the source, follows much the same course as in Art. 9. The full 

 interpretation is however more difficult, so far at least as regards the minor tremors. 



The main part of the disturbance, in the case of a local vertical pressure applied to 

 the surface, is obtained by generalizing the formulas (159). These may be written 



o = H - K-- \*e i (t - a>eo *">du + &c., w,= - lK n - J fV ( "- OMh >dw + &c. (162). 



TT fJL Cm Jo TT fj. Ct Jo 



Hence, corresponding to an arbitrary pressure R (t), we have 



/ = H R( cts cosh u) C/M+&C., iv (} =- LT( CCTCOsh ) c/-|-&c. (163), 



ff/A OwJo 7T/X Ot Jo 



where, in analogy with (100), 



K (t) = - 1 - f dp ( R (X) ship (t -X)dk ..... (1G4). 



77 JO J -* 



The character of the function oft represented by the first definite integral in (163) 

 has been examined by the author t for various simple forms of R (t), and a similar 

 treatment applies to the second integral. For example, if we take 



it is found, on putting 



t COT = T tan x, 



that for values of ro large compared with r/c, and for moderate values of x, 



Til (t - cv, cosh M) (^ = R A/( 2T ) cos O - k) v/(co x) ( ' ^)t 



Jo fiT " \CTSj 



I ' K (< - rw cosh u) dn = - ^ \/( 2T ) sin (iff - -J X ) v/'( cos x) ( L67 )> 

 Jo 2r v \CBT/ 



approximately. Substituting in (163), we have, ignoring the residual terms, 



'/o = -/i" (^ ~ tx) cos' 

 ">u = .'/ cos (** tx) cos ' X 



* Cf. the footnote on p. 2 ante. 



t "On Wave-Propagation in Two Dimensions," ' Proc. Loud. Math. Soc.,' vol. 35, p. 141 (1902). 

 t cy. Equation (36) of the piper cited. It may Ije noticed that the functions on the right hand of (166) 

 and (167) are interchanged, with a change of sign, when we reverse the sign of x- 

 The symbol x is no longer required in the sense of equations (115), &c. 



