1885.] 



Mr Ibbetson, On an elastic shell. 



313 



For obvious reasons this error has no effect on the apparent 

 verification of the former results, by reduction to the case of con- 

 centric spheres. 



The correction greatly simplifies the expressions for the 

 potential energy, and they may now be integrated with com- 

 parative ease. 



With our former notation we have, for the oblate shell 



T _ ir pa/3 2 (3 + 4a 2 + 8a 4 ) £,") 

 1 ~60 " ~ ^T~ 



W = 



q/3 6 A 



> , 



1 192 ' (1- yU/W 



and for the prolate shell 



- £' 2 



7T pq/ y(8+4a' + 3a') . ] 

 2 * "60 ' a 2 " ? 



F = 



<?/3 6 ^ 3 



k 



192 * (l-/**)aa° 



1 rcos~ia 



where J = -7== {9 cos 4 - 6 [1 + (1 - /*) a 2 ] cos 2 



Jl -a 2 Jo 



+ 1 + 2 (1 - 4/x) a 2 + 2 (1 + 2/a) a 4 



+ 2a 2 (1 - a 2 ) [> - (1 - fi) a 2 ] sec 2 



+ a 4 (1 - a 2 ) 2 sec 4 6) d$ ; 



I f C0S~1 a 



A = 



(9a 8 sec 5 5 - 6a 6 (1 - fi + a 2 ) sec 3 



Jl-a'Jo 



+ a 4 [2 (1 + 2fi) + 2 (1 - 4/a) a 2 + a 4 ] sec 

 + 2a 2 (l-a 2 )(l-,u-/*a 2 )cos0 

 + (1 - a 2 ) 2 cos 3 0} e£0. 

 Thus, on integration, 

 A 1 = (tf + 3*0 a- (V -3/*)a 3 + (1 - 2/i) a 5 +f a 7 



+ 



cos a 



li 



(1 + 5/*) a 2 + 2 (1 + 2/*) a 4 ] 



