1888.] Bending and Vibration of thin Elastic Shells. 117 



ness which, if variable at all, is a function of z only. Since u t v, w 

 are periodic when increases by 27r, their most general expression in 

 accordance with (31) is [compare (23), &c.] 



w = 2[* (A 



u = 2[ ^Bja sins0 s^B/a coss0] 



cos 50 (A s 'a + B s 'z) sin$0] ...... (35), 



sin s0 + s (A/a+B/0) cos s0] ---- (36), 



(37), 



in which the summation extends to all integral values of s from 

 to oo. But the displacements corresponding to s = 0, s = 1 are 

 such as a rigid body might undergo, and involve no absorption of 

 energy. When the values of u, v, w are substituted in (34) all the 

 terms containing products of sines or cosines with different values of 

 s vanish in the integration with respect to 0, as do also those which 

 contain cos s0 sin s0. Accordingly 





P m 1 



- -- s 

 [_m + n a? 



2 s 3 



.. (38). 



Thus far we might consider A- to be a function of z ; but we will now 

 treat it as a constant. In the integration with respect to z the odd 

 powers of z will disappear, and we get as the energy of the whole 

 cylinder of radius a, length 2Z, and thickness 2h, 



+1&* 



r^'f** 

 = 1 ( TTa 



J-Jn 



in which s = 2, 3, 4, 



+ B, a +B,'] (39), 



The expression (39) for the potential energy suffices for the solu- 

 tion of statical problems. As an example we will suppose that the 

 cylinder is compressed along a diameter by equal forces F, applied at 

 the points z = z lt = 0, = TT, although it is true that so highly 

 localised a force hardly comes within the scope of the investigation in 

 consequence of the stretchings of the middle surface, which will 



i 2 



