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



71 



Hence if p be the density of the shell, and T the kinetic 



p £ 2 

 energy of the motion, we get by writing U=~. -rn. • V • T i n the 



& T?ft C 



integration formulas (2) and (3) 



ir P 6(a i +2c i ) l 2 

 12 . aY ' ? ' 



whether the shell be oblate or prolate. 



Again, if v be the potential energy of the strain per unit of 

 unstrained surface, we know that 



where q is " Young's Modulus," [x the ratio of lateral contraction to 



longitudinal extension, and 8 — , 8 — are the small increments of the 



Pi P*. 



principal curvatures due to the strain. 



Hence, in our case, 



qr 3 £ 2 (Yd 1 



v = 



+ 



SI 



v = 



24 (l-/, 2 )' IL^ftJ 

 , and ] 



{v 2 [v(a 2 -c 2 )-aY] 



+ 2p 



3^J 





Differentiating (1), and putting £ = 0, 



96(l-yu, 2 )a 6 cV 



+ a 4 [77 (a 2 + c 2 ) - 3a 2 c 2 ] 2 + 2fta* V [a 4 (77 - 2c 2 ) 2 - c 4 (77 - a 2 ) 2 ]}. 



Hence if Tf 1} TP 2 be the integrals of the potential energy for 

 the prolate and oblate shells, we get by putting U— v in (2) and 

 (3) successively, 



dv 



+ a 4 [ v (a 2 + c 2 ) - 3a 2 c 2 ] 2 + 2fm 2 V [a 4 (77 - 2c 2 ) 2 - c 4 (17 - a 2 ) 2 ]} 



{7?(a 2 + c 2 )-aV} 3 



rfJv-C* 



W = 



■qeT 



= fV[^ 2 -c 2 )-a 2 c 2 ] 2 

 — « i» s 



384 (1 - ,0 a 8 cVc 2 - a' 

 + a 4 fr (a 2 + c 2 ) - 3a 2 c 2 ] 2 + 2^77 [a 4 (77 - 2c 2 ) 2 - c 4 (77 - a 2 ) 2 ]} 



{>? (a 2 + c 2 ) - a 2 c 



CZ77 



ifjc*-rj' 



