500 EEPORT— 1889. 



term due to stretching is negligible in comparison with the former. Mr. Love, on 

 the other hand, considers that the term due to stretching is of greater, or at any 

 rate of equal importance to that due to bending, and also that it is impossible 

 to satisfy the conditions at a free edge, if the shell ■\ibrates in such a manner 

 that every line on the middle surface is unaltered in length. 



For the purpose of examining the theory proposed by Mr. Love, I think it will 

 be best to avoid for the present making any assumption as to the relative im- 

 portance of the two terms of the potential energy, and to retain them both ; and I 

 shall therefore in the first place consider the equilibrium of an indefinitely long 

 cylindrical trough fiUed with liquid of density p, whose cross section is a semi- 

 circle of radius a, and which is supported by vertical strings attached to its 

 edges. 



Putting 



Tfi _ 87nnh 

 {'in + n)a 



I find that the extension is constant and equal to <7pn-a/4E, and that the change 

 of curvature is 



Jr^WL^-^-^^''^-'-^^ (^> 



where (f> is measured from either of the edges. We therefore see that the change 

 of curvature vanishes at both the edges. 



If Wj be the potential energy per unit of area due to bending, and W^ the 

 corresponding quantity due to stretching. 



Since h is small in comparison with a, we see that Wj is large compared with 

 W„, except in the neighbourhood of the two edges. 



In the preceding problem it is evident that the middle surface must be sub- 

 jected to a considerable strain, owing to the fact that the shell has to support the 

 weight of the whole of the liquid which it contains ; and it seems hardly probable 

 that when vibrations are set up by striking a cylindrical shell a greater extension 

 of the middle surface is thereby produced than in the statical problem just con- 

 sidered, I therefore think we are justified in concluding that in the case of metal 

 shells the bending term is the most important. It would not, however, be safe to 

 apply this conclusion without further investigation of the case of indiarubber 

 shells, under the influence of external pressure, inasmuch as the potential energy 

 due to the contraction of the thickness might have to be taken account of. 



Mr. Love has also raised the objection that when a cylinder is vibrating it is 

 impossible to satisfy the conditions at a free edge if the middle surface is supposed 

 to be inextensible. We shall now, for the purpose of examining this objection, 

 consider the vibrations of an indefinitely long circular cylinder, when the dis- 

 placement of every point lies in a plane perpendicular to the axis. 



If we omit the stretching terms and also the terms due to the rotatory inertia, 

 the equation of motion is 



where 



_\ _ d^v di\ 

 d(fi^ d(f)^ 



and if the cross section of the cylinder is an arc of a circle of length 2aa, the 

 conditions to be satisfied at the two generating lines which form the edges of the 

 shell are 



X = (4) 



^ = (5) 



