10 PEOFESSOR HORACE LAMB ON THE PROPAGATION OF 



The surface-values of the displacements are now given by (33), viz. : 



P - 2a ** Y 



_ 

 : 



Y 



The effect of a concentrated force Q acting parallel to y at points of the line 

 x = 0, y = is deduced, as before, by writing Y = Qd/2ir, and integrating 

 from oo to oo ; thus 



tQ f g_(2_e - V-Jtap)^ 

 ~ ~2iL. ~F(f) ~ 



f ..... (52). 

 Q f" PaeF^dg 



~2vJ-. F"f 



In a similar manner, corresponding to the tangential surface forces : 



[f.ry]o = Xe' f *, \_pyJ\Q = (53), 



we should find 



XO f-2 7,2 Y 

 T> ^JC /^ -^X 



And, for the effect of a concentrated force P acting parallel to x at the origin, 



P 



"2ir^-. 



h ..... (55). 

 iP p ^(2^-F-2^)^c^ I 



2/tJ- ~'F"(f)"" J 



The comparison of ?? in (52) with - in (55) gives an example of the general 

 principle of reciprocity.* 



We may also consider the case of an internal source of disturbance, resident 

 (say) in the line x = 0, y=f, the boundary y being now entirely free. The 

 simplest type of source is one which would produce symmetrical radial motion (in 

 two dimensions) in an unlimited solid, say 



< = D (hr), V = . . ....... (56), 



where r, = ^/{x + (y /) 3 j, denotes distance from the source. If we superpose on 

 this an equal source in the line x = 0, y = f, we obtain 



< = DO (/) + D (Ar'), ,/, = o ..... . . (57), 



* RAYLEIGH, 'Theory of Sound,' vol. 1, 108. 



