72 Mr Ibbetson, On the small free normal vibrations [Jan. 28, 



It will be convenient at this stage to make the a and c of the 

 prolate shell respectively equal to the c and a of the oblate shell. 

 These symbols will then denote in all the formulae the major (a) 

 and minor (c) semiaxes of the meridional section common to the 

 two shells. 



Thus, interchanging a and c in the prolate formulae, and 

 writing 



we have for the oblate shell 



T= i L pa/3 2 (1 + 2a 4 ) ' 



1 1 «> • a 3 ' S 



w = 



12 



7T 



1 12" 16(1 -/a 2 ) aoi 1 

 and for the prolate shell 



*■ pa/3 2 (2 + a 4 ) 4, 



2 12 " ~ 2 



* , 



F=^ 



a 



12 '16(l-/A 8 )aa 8 



r , 



where A 1 and A 2 are functions of a only, which we may write 

 in the form 



"1 /"COS* 1 a 



A t = {a 4 (sin 2 6 - a 2 ) 2 + cos 4 6 (1 4- a 2 - 3 cos 2 Of 



VI — a Jo 



+ 2^ cos 2 [cos 2 20 - (a 2 - cos 2 6>) 2 ]} (sin 2 + a 2 ) 3 . cos 2 6 . d6>. 



1 /"COS -1 a 



A = ■ . (cos 4 6> (cos 2 (9 + a 2 sin 2 0) 2 + a 4 [( 1 + a 2 ) cos 2 - 3a 2 ] 2 



vl — aVo 



+ 2^a 2 cos 2 6 [a 4 (1 + sin 2 0) 2 - (cos 2 6 - a 2 ) 2 ]} 

 (cos 2 0-a 2 sin 2 6>) 3 



dO 



cos 17 



These integrations are easily performed, term by term, though 

 the calculation of the numerical coefficients is tedious. 



The results may be written 

 A 1 = h +h 1 a 2 + ... + h 7 <x li + (k + k 1 a 2 + ... + k 7 * u ) 



cos * a 



vr^r 2 ' 



,o g (I±^Z) 



A =m + m 1 <x 2 + . . . + ra 7 a 14 + (« + n^a 2 + . . . + n 7 a u ) 7=== 



vl — a 



