13:2/ Mechanical Resonances of Biological Cells 239 



where T is the interfacial tension. Differentiating Equation 8 with 

 respect to time and recognizing that 



8R 8<p 



-— = — - - at r — a 

 dt 8r 



one finds 



dt dt a 2 sin 6 dd \ drddj 



T 8 3 cpi 2 T d<p { 



+ 2 • 2fl F^Tl ~ — '-£ at T = a 9 



ar sin 2 6 drdifj^ a z or 



The problem, then, is to find a solution to the Equation of continuity 

 (2') and the Equations of motion (4') obeying the boundary conditions 

 (5), (6), (7), and (9). 



Because the problem has been set up,in spherical coordinates, the most 

 general solutions to Equation 2' are 



<p t = ^A n r-e-^S n {e^) 



" = oo° ( 10 ) 



n = 



where A, B, and a> are constants, j the square root of - 1 , and 



S n (d, «A) = 2 ^n (COS 6) *" 

 m = 



In this, the a's are constants and P™ is the mth associated Legendre 

 polynomial of order n. It may be readily shown that 



1 a / . a dS n \ 1 d* _ n(n + 1) 



It is possible to satisfy the boundary conditions only if the frequency 

 has certain discrete values given by 



,2 



T n{n - \){n + 2) 



«:--,^ . - . . (11) 



" /V* 3 



1 + 



[n + 1/ p. 



For the lowest possible mode, n = 2, this becomes for Pi = Po = 1, 



a>i = 4.87a- 3 (12) 



This formula has been used to estimate some of the interfacial tensions 

 referred to earlier. When a higher harmonic is used, the values 

 obtained for Tare lower than those computed from Equation 12. 



