444 Lord Rayleigh. Note on the Free [Mar. 14, 



The functions w, v, w are to be assumed proportional to the sines, or 

 cosines, of pa and 50. These may be combined in various ways, but 

 a sufficient example is 



u = U cos s0 cos JLLZ, v = Y sin 50 sin #0, w = W cos 50 sin pa. . (3) ; 



so that a\ = /tU cos s0 sin ^z ................. f (4), 



<r 2 = (W + aV) cos s0 sin ^ ............ . . (5), 



w = ( sU+ynY) sin s0 cos yuz ............ (6), 



unity being written for convenience in place of a. The energy per 

 unit area is thus 



m+n 



+ sin 2 s0 costpz (-sU+/tV) 2 l ..... (7). 



Again, the kinetic energy per unit area is, if p be the volume 

 density, 



ph Y~^Y cos 2 50 cos 2 fas + Ysin 2 S0 sin 2 pa + Y cos 2 50 sin 2 



(8). 



In the integration of these expressions with respect to and z, the 

 mean value of each sin 2 or cos 2 is J.* We may then apply Lagrange's 

 method. If the type of vibration be cos pt^ and p^p/n = & 2 , the re- 

 sulting equations may be written 



= 0. . . . (9), 

 = 0... (10), 

 = 0... (11), 



where M = ^LT? . . (12). 



The frequency equation is that expressing the evanescence of the 

 determinant of this triad of equations. 



We will .consider for a moment the simple case which arises when 

 /i = 0, that is, when the displacements are independent of z. The 

 three equations reduce to 



* In the physical problem the range of integration for $ is from to 2ir ; but 

 mathematically we are not confined to one revolution. We may conceive the shell 

 to consist of several superposed convolutions, and then s is not limited to be a whole 

 number. 



