540 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS 



Calculating from these, we find, dropping the time-factor, 



erj = cos s<j> [ftjPj sin n^z + /AC PC sin p.. z z + f^P^ cos p. t z + f^P^ cos /A 2 z], 



K, \ / s \ -i 



/*iQi '- p i cos /^z + (/i 2 Q a -- P 2 cos /tgz 

 I \ / J 



sin 6-< I //* 1 Q / , P'j sin /t^ + 2 Q' 2 - P' 2 sin 



J 



If there is a rigid disc at z = 0, then v and w vanish with z, so that 



........ (100) 



The first of these is, by (96), 



\ + 



a T- 

 /tf D 



so that (100) can only be satisfied by P' 1? P 2 , both zero, and consequently Q'j, Q' 2 , 

 both zero, unless we take /u, t 2 = /i 3 2 and Q\ + Q' 2 = 0. 



If pf = /i 2 2 , we have P^ = ^ P' 3 and Q\ = Q' 2 > so that the terms in u, v, w, 

 o-j, cr 2 , IB- which contain P'j, P' 2) Q' l5 Q' 2 all vanish identically. 



It follows that to satisfy the conditions at z = we must drop out the P', Q' terms. 



The boundary-conditions at ' = c are 



^ = 0* 

 where we have to take only the part in Pj, P 2 , Q 1; Q 3 and to write 



Mi D M.>* I) 



Hence, we have 



a* <rs8 + l C 1 . , c* <rs8 + l C 



and 



