462 



SURFACE WAVES. 



[CHAP. IX 



of Solid Geometry, that if X, p, v be the direction-cosines of the 

 normal at any point of a surface F(x, y, z) = 0, viz. 



1 1 d\ diji dv 

 then ~p~&quot;*~fc r = ;7~&quot;*~77 + ;7~ ( ) 



Since the square of ? is to be neglected, the equation (1) of the 

 harmonic spheroid may also be written 



r = o + ft,, (6), 



r n 



where f ?l = S n . sin (at + e) (7), 



CL 



i.e. % n is a solid harmonic of degree n. We thus find 



X d% n X y 



f dii w& V / 



Z Cv^n ^ ^ 



~r~dz l r 2 ^ n 

 whence 



~~ a a 2 



Substituting from (4) and (9) in the general surface-condition of 

 Art. 244, we find 



If we put p = 0, this gives 



The most important mode of vibration is the ellipsoidal one, 

 for which n = 2 ; we then have 



