544 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS 



assumed by English writers that the reverse of this is the case, viz., that the vibrations 

 take place in such a way that no line on the middle-surface is altered in length. This 

 will be borne out by a reference to Lord KAYLEIGH and IBBETSON. The theory of 

 the present paper rests on the fact that the functions expressing the stretching and 

 those expressing the changes in magnitude and direction of curvature are of the 

 same order of small quantities. This is proved in the following way : The potential 

 energy consists of two parts ; one, Q 2 , proportional to the thickness h ; and the other, 

 Qj, proportional to h s . The first is expressed in terms of the stretching, and the 

 second in terms of the bending of the middle-surface. Some previous theories have 

 proceeded as if Q l only occurred. If this were the case, we ought to get an approxi- 

 mation by supposing that Q%/h = 0. This is equivalent to assuming that there is no 

 stretching of the middle-surface. We should therefore get an approximation by 

 supposing the surface inextensible to the first order. The stretching and the bending 

 are expressed, to the first order, by linear functions of certain differential coefficients 

 of the displacements. Our supposed method of getting an approximation is then to 

 make the functions expressing the stretching vanish. Now, I have shown that the 

 functions expressing the displacement are thus, to a certain extent, determined, and 

 that iu such a way that the boundary-conditions cannot be satisfied. The boundary- 

 conditions referred to are the exact conditions found by retaining the complete 

 expression for the potential energy. Tt is inferred that the functions expressing the 

 stretching cannot be taken equal to zero for an approximation ; or, in other words, 

 small compared with those expressing the bending ; and, thus, QJh 3 and Q 2 /7i, are of 

 the same order of magnitude. The conclusion that Qj is small compared with Q 2 

 seems inevitable. 



The argument breaks down for a plane plate through the vanishing of the curvatures ; 

 Q l is then alone of importance. In the case of an open shell or bowl whose linear 

 dimension is small compared with its radius of curvature, and large compared with its 

 thickness, both terms are important. When this is so, we get a class of cases for 

 which the linear dimensions concerned are of three different orders of magnitude, and 

 this case will not come under the method of the present paper. It may be compared 

 with the problem of the watch-spring mentioned in THOMSON and TAIT'S ' Natural 

 Philosophy,' Part 2, p. 264, which stands between a bar and a plate. The very open 

 shell or bowl stands in the same way between a plate and what I have called a shell. 



The theory of this paper proceeds as if Q 2 alone occurred. It is to be regarded as 

 the limiting form for indefinitely thin shells. A complete theory of bells, even when 

 regarded as uniformly thick and isotropic, could only be obtained by using the exact 

 equations formed by retaining both terms of the potential energy. 



Again, English writers have assumed that the potential energy, which they suppose 

 to depend only on the bending, will be the same quadratic function of the changes of 

 principal curvature as it is for a plane plate. The same authorities as before may 

 be quoted, and we may also refer to a question set in the Mathematical Tripos, 



