TREMORS OVER THE SURFACE OF AN ELASTIC SOLID. 

 and therefore, for z > 0, 



31 



(127). 



To pass to the case of a concentrated force Re 1 '', acting parallel to z at the origin, 

 we have recourse to the formula (20), where we suppose f(\) to vanish for all but 

 infinitesimal values of X, and to become infinite for these in such a way that 



f" 



J 

 Jo 



2ir\d\ = R. 



We therefore write Z = Rf d/2Tr, and integrate with respect to f from to QO.* 

 We thus find, for z > 0, 



- ' 28) ' 



which are equivalent, by (18), to 



R 3 e~* 



-- . 



' Bz r 



Y 



R 



. 

 4:Trp 2 p ' r 



(129) 



This will be found to agree with the known solution of the problem.! If we retain 

 only the terms which are most important at a great distance r, we find, from (1 18), 



/7 

 " " " 



R f 1 



- J 

 " " 



4 77 [ X + 2u r 3 



,,.3 



R J 



IV = . 4 : 



4irU + 2 /i r s p. 



Inserting the time-factor, the radial displacement is 



(130). 



ZW 



R 



and the transverse displacement in the meridian plane is 



zq _ R CT >( ,_ M 



- o c/ 



Returning to the exact formulae (128), the expression for the velocity parallel to z 

 at the plane z = is found to be 



o (,)# ...... (133), 



* A more rigorous procedure would be to suppose in the first instance that the force R is uniformly 

 distributed over a circular area of radius a, using the formula (22). If in the end we make a = 0, we 

 obtain the results in the text. 



t STOKES, 'Camb. Trans.,' vol. 9 (1849); ' Mathematical and Physical Papers,' vol. 2, p. 278. 



