1884.] of a thin homogeneous and isotropic elastic shell. 



73 



The equations of free vibration of the two shells become 

 q PA* ^ 



1 + 

 £ + 



16p(l-^ 2 )'(l + 2a 4 )a 9 a 

 q &A 



16 p (1 - /a") ' (2 + a 4 ) a 6 a ! 

 and if t t , t 2 be the periodic times, 



F.f = 



h , 



_87ra /pOjzHl) ^0- + ^) ) 

 "V" a " -4. 



/3 2 



_ 87ra / 

 - -ar V 



P(l-/^ 2 ) ^ (2 + a 4 ) 



/3 2 V .gr ' A, 



Thus the times of vibration of similar shells of the same kind 

 are in the simple ratio of their linear dimensions. 



As a verification of these results we may deduce from either 

 the case of a thin spherical shell of radius a and uniform thickness 

 t, performing small radial vibrations so as always to retain the 

 spherical form. 



We are to put 



a = l, /3 2 = 



2r 



Now A x and A 2 are both of the form 



Vl 



"1 r cos-la 



L== F {a, 9) old, 



-a 2 Jo 



and when a=l, this becomes by evaluation F (1, 0), which 

 = 2(1+ /j,) in both cases. 



Hence our formulae reduce to 



Now, with our previous notation, if a + u be the radius of 

 the spherical shell at time t, 



T=\ 9 Atto?.t.u\ 



and 



v = 



q r 



24 (1 - p, 2 ) 

 .'. F = 4tto 2 . 



•2(1+ a*) 



a + u 



2 



. ? 



M = 



OT U 



