TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 13 



and (30). We merely note, for purposes of reference, that if in (33) we put = i K , 

 and accordingly, from (61), 



A = (2 K " - F) C, B = 2ifc 1 C ...... (64), 



we obtain by superposition a system of standing waves in which 



u = 2* (2/r k* 2^) C sin KX . e*', v = 2& 2 a 1 C cos KX . e ! f { . (65). 



The theory here recapitulated indicates the method to be pursued in treating the 

 definite integrals of Art. 5. We fix our attention, in the first instance, on their 

 " principal values," in CAUCHY'S sense, and afterwards superpose such a system of free 

 Rayleigh waves as will make the final result consist solely of waves travelling 

 outwards from the origin of disturbance. 



It may be remarked that an alternative procedure is possible, in which even 

 temporary indeterminateness is avoided. This consists in inserting in the equations 

 of motion (23) frictional terms proportional to the velocities, and finally making the 

 coefficients of these terms vanish. This method has some advantages, especially as 

 regards the positions of the "singular points" to be referred to. The chief problem 

 of this paper was, in fact, first worked through in this manner ; but as the method 

 seemed rather troublesome to expound as regards some points of detail, it was 

 abandoned in favour of that explained above. 



7. The most important case, and the one here chiefly considered, is that ot a 

 concentrated vertical force applied to the surface, to Avhich the formulae (52) relate. 

 The case of a horizontal force, expressed by the formulas (55), could be treated in an 

 exactly similar manner. 



Since w is evidently an odd, and v () an even, function of x, it will be sufficient to 

 take the case of x positive. 



As regards the horizontal* displacement it () , we consider the integral 



L ({) d r = 

 J 



3 - A 8 ) 



taken round a suitable contour in the plane of the complex variable , = f + *V 

 If this contour does not include either "poles" ( *, 0), or "branch-points" 

 ( h> 0)) ( & 0) of the function to be integrated, the result will be zero. 



A convenient contour for our purpose is a rectangle, one side of which consists of 

 the axis of except for small semicircular indentations surrounding the singular 

 points specified, whilst the remaining sides are at an infinite distance on the side 77 >0. 

 It is easily seen that the parts of the integral due to these infinitely distant sides 

 will vanish of themselves. If we adopt for the radicals ^/(Z,* h") and v/( a & 2 ), 



* The sense in which the terns "horizontal" and "vertical" arc used is indicated in the second 



sentence of the Introduction. 



