32 PROFESSOR HORACE LAMB ON THE PROPAGATION OF 



or, taking the real part, 



47TP/3 



+ terms in sin pt 



The terms in cos pt remain finite when we put ts = ;* and the mean rate W at 

 which a force R cos pt. does work in generating waves is thus found to be 



W = 



Jo 



(a 3 + 2fc 3 ) . (135), 



24ir/o V 



a and ?> denoting as before the two elastic wave-slownesses. The result (135) can be 

 deduced, as a particular case, from formulae given by Lord KELVIN, t 



12. Proceeding to the case of a semi-infinite solid occupying (say) the region z> 0, 

 we begin with the special distribution of surface-stress : 



[>] (I = Z. J (1 (&0, [>-] = ....... (136). 



The coefficients A, B in (119) are now determined by 



Z ] 



** ^ ....... (137), 



2A - (2^ 2 - jfe 9 ) B = J 



whence 



A _2P-/^ Z B _ 2 Z n . 



A ~ FffT'7' *X) 7 ...... (38)> 



the function F (^) having the same meaning as in Art. 5. The corresponding 

 surface-displacements are 



. Z 



F () ? 



}> (139). 



Fa ,, x Z 



'o = -p () ' (' ' 



This result might have been deduced immediately from (51) in the manner indicated 

 at the beginning of Art. 10. 



t The terms in sin pt become infinite. If the force R be distributed over a circular area, the awkwardness 

 is avoided. A factor 



iteOV 

 ifo J 



is thus introduced under the integral signs in the first line of (135), where a denotes (for the moment) the 

 radius of the circle. Finally, we can make a infinitely small, 

 t 'Phil. Mag.,' Aug. 1899, pp. 234, 235. 



