18G 
MK. L. N. G. FILON ON THE ELASTIC EQUILIBRIUM OF 
exj^aiiding is not adequate, and that to obtain any real effect, we require to make 
the constraining rim of a certain definite thickness. 
In so doing, we are reaUy introducing an additional condition, besides the non- 
exj^ansion of the ends, the cylinder being now, as it were, built-in. The jDrohlem as 
it stands did not appear of sufficient interest to warrant the expenditure of arith¬ 
metical labour upon it, so I have contented myself with stating the algebraical 
results. 
§ 13. The Second Problem: Constraint effected hy Shear over the Terminal 
Cross-sections, Determination of the Constants. 
Supjjose now that we consider our cylinder subject to the following conditions :— 
(i.) A total pressure ircd over the plane ends, the distribution of this pressure 
being unknown. 
(ii.) The ends constrained to remain plane, so that iv — const, when 2 ; = c. 
(hi.) The ends not to expand alojig the jjerimeter 
u = 0 when 7 - = a, z = ff: c. 
This condition is satisfied by allowing a shear rz over the plane ends, its 
distribution being, however, unknown. 
(iv.) No stress across the curved surface, i.e., 
rr =. 0 when r = a, 
rz = 0 when r = a. 
These conditions will represent the state of things which we may expect to hold it 
the cylinder be compressed between two rigid })lanes which are sufficiently rough to 
prevent the expansion of the ends. 
Now, in such a case as this, it is obvious that the exj^ressions for the stresses and 
strains as purely periodic series in z break down, for if we take the expressions (24) 
and (26) for 11 ' and rz the condition that iv = const, when z = ^ c will give us, as 
before, E = 0, and the vanishing of the stresses at the curved surface will give two 
homogeneous equations of condition between A^, Ao, and C. These, taken in con- 
iunction with equation (35), give three linear homogeneous equations in A^, A^^, and 
C, which are in) general inconsistent unless Aj = 0, Ao = 0, C = 0, which would 
destroy the periodic solution altogether. 
We have therefore to assume that u and tv are made up ol two parts. The 
first part, which I shall denote hy U, W, consists of the periodic solution hitherto 
obtained. The second part is a finite power series in r and z. The resulting expres¬ 
sion is a combination of the two ty})es of solution, which are discussed separately hy 
Mr. Chree (‘ Camh. Phil. Soc. Trans.,’ vol. 14). Either of these two types, taken by 
