Operative at one or more Points of an Elastic Solid. 301 

 The resolved displacement is 



7 cos (/) -f a sill (^, , (26) 



a and 7 being given by (18), (19), in which we write 



~/r = co^0, a;lr = smO. 

 We lind 



7 cos (j)-j-a sin (p 



= cos ipt-kr) [-sin 9 sin 0' -""^-^ + p^^^^os 6> cos ^'] 



sin (yjf— /I'r) r , o n n,-x 



+ - ^^^-^^ [cos (f)-3 COS 6* cos 6'^ 



kr 



a symmetrical function of 6 and 6' as required by the general 

 principle of reciprocity ('Theory of Sound/ § 108). The 

 value of A is given by (17) in which, however, we will now 

 write ^ for Zj, so that 



^=Jp, (2^) 



The above equation gives the resolved displacement at P 

 in a direction making an angle 6' with r due to a force 

 §cos^:>^ at acting in a direction inclined to r at angle 6. 

 If we suppose that a force 5' cos />^ acts at P indirection 

 PT and inquire as to the work done by this force upon the 

 motion due to §•, we have to retain that part of the resolved 

 displacement due to § which is in quadrature with ^^cosj?^, 

 viz. the part proportional to ^irv pt. The mean work is given 

 by the symmetrical expression 



F^^-^ /cos [6^-6') -a c os 6' cos 6 ' ( , sin kr \ 



-^%in^sin^^| (28) 



If the forces are parallel, c/) = 0, 6'=— 6, and (28) 

 becomes 



^^W^' V • 9 /J sin At , 1 — 3cos2^/ sin AtXI 



If we further suppose that kr is very small, the square 

 bracket reduces to the value ^, and w^e get 



Vlirhp ^^^> 



