TRANSACTIONS OF SECTION A. 501 



-lSali^'^^)->'''d^^'' '^^^ 



The total stress across any section made by a plane passing through a gene- 

 rating line consists of a couple, a normal shearing stress, and a tension perpen- 

 dicular to the generating line ; ' and equations (4), (5), and (G) are the conditions 

 that these stresses should vanish at a free edge. 



In order to solve these equations, we must assume that v varies as (^(■'**p'>. 

 Substituting in (3), we find that the relation between « and p is 



n_ 4nmh-s-(s" — l)- ^»v 



'6{m + }i)(s- + l)pa* 



The value of s in terms of p are the six roots of (7), but in order to obtaiii 

 the frequency, the value of s in terms of the dimensions and elastic constants is 

 required. Now each of the boundary conditions must be satisfied at each of the 

 two edges of the shell, and therefore there are altogether six equations of con- 

 dition. Hence the six constants which appear in the solution of (3) can be 

 eliminated, and the resulting determinantal equation combined with (7) will give 

 the frequency. 



If the cylinder is complete, s is an integer, and the frequency is determined by 

 (7), which is the result obtained by Iloppe and Lord Rayleigh. 



If a cylindrical shell of finite length is bent along a generating line in such a 

 manner that its curvature is increased, all lines parallel to the axis which lie on the 

 convex side of the middle surfiice will be contracted, whilst all such lines which lie 

 on the concave side will be extended, and this contraction and extension will give 

 rise to a couple about the circular sections which tends to produce anticlastic 

 curvature of the generating lines. One of the boundary conditions requires that this 

 couple should vanish at the two circular edges ; and consequently the preceding 

 investigation does not apply when the cylinder is of finite length. 



6. Simplified Proofs (after Euler) of the Binomial Theorem (i.) for any 

 Positive Fractional Exponent; (ii.) for any Nerjative Exponent. By 

 T. Woodcock, M.A. 



(i.) We assume the theorem proved for any positive integral exponent. 

 Let h and k be positive integers throughout, and let .v be numerically less 

 than unit}', so that the series occui-riug are convergent. 

 Since 



{(l+.r)"'}^ = (l+.r)»'»--; 



therefore the ^th power of 



1 + mo.- + -Aj-^ — Ar + . . . . 



is always 



1 + mkx + — '■ — —r- + . . . . 



"when m is a positive integer ; therefore, also, when hi is a positive fraction [since 

 the form of the said /cth power cannot be affected by >;» being integral or fractional]. 



Let j/i = — , then mk = h ; 

 k' ' 



that is (1 + x)'', since h is a positive integer. 



' Compare Besant, ' On the Equilibrium of a Bent Lamina,' Quart. Journ., 1860. 



