462 SURFACE WAVES. [CHAP. IX 



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

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



1 1 d\ da dv 



then -j?+jy = j- + ir + j- .................. ( 5 )- 



R! jR 2 dx dy dz 



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

 harmonic spheroid may also be written 



, ........................... (6), 



where n = ^ S n . sin (at + e) .................... (7), 



i.e. f n is a sofo'c? harmonic of degree 71. We thus find 



T dz T 2 



whence 







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

 Art. 244, we find 



If we put // = 0, this gives 



i ................ (11). 



The most important mode of vibration is the ellipsoidal one, 

 for which n = 2 ; we then have 



