8 PROFESSOR HORACE LAMB ON THE PROPAGATION OF 



Hence 



A = - A ' = ^- B = B ' = f'4 ' ' ' ' ' (38> - 



We have, then, for y > 0, 



" -* 



To pass to the case of an extraneous force Q concentrated on the line x = 0, y = 0, 

 we make use of (19). Assuming that the /(X) of this formula vanishes for all but 

 infinitesimal values of X, for which it becomes infinite in such a way that 



we write, in (39), Y = Qf//2ir, and integrate with respect to f from oo to + co.* 

 We thus obtain, for y > 0, 



, Q r e~v#, + = - f &:** . . . 



47T/C-/A J -x 47T&->t J -oc p 



or, on reference to (13), 



where r = \/(% 2 + ?/ 2 ). 



If we put x = r cos 6, y = r sin #, we find from (25), on inserting the time-factor, 

 that for large values of r the radial and transverse displacements are 



a< ai// _._ Q v/JL e ; ( /'-*-- to s ; n fl 



r " - 



cos ^ 



respectively.! Use has here been made of (7). 



A simple expression can be obtained for the rate (W, say) at which the extraneous 



* The indeterminateness of the formula (19) in this case may be evaded by supposing, in the first 

 instance, that the force Q, instead of being concentrated on the line .< = 0, is uniformly distributed over 

 the portion of the plane ,'/ = lying between x= a. It appears from (21) that we should then have 



Y= Q in& d c 

 2?r a 



If we finally make a = we obtain the results (40). 



t The second of these results is equivalent to that given by RAYLEIGH, loc. cit., for the case of 

 incompressibility (A. = oo ). 



