154 
MR. L. N. G. FILOX ON AN APPROXIMATE SOLUTION FOR BENDING A 
sections distorted. As a matter of fact, what really happens at a built-in end is 
quite unknown. Under these conditions any solution which makes U = 0, cWldx = Q 
over the ends must be restricted to the case of an infinite continuous beam resting 
upon a series of equidistant supports, each at the same vertical height; the load 
carried by the beam being exactly rej)eated over each span. A rail under its own 
weight and carried on sleepers is an approximate example. In this case Pochhaw^ier 
and Eibiere’s solutions are exact, and it is then legitimate to make the span as small 
as we please. 
In practice such conditions will but rarely occur, because, as is well known, any 
slight difference in the height of the supports, or in the manner in which the beam 
bears upon them, will upset the symmetry altogether. 
The ultimate stej^ in the process of thinning down the boundary conditions is taken 
when one of the two boundaries of the infinite jolate is itself removed to infinity, 
leaving only one plane bounding an otherwise unlimited solid. 
This problem also has been solved by Lame and Clapeyeox {loc. cit.) in terms of 
quadruple integrals. In this case the limiting conditions at infinity cease to be 
important, because, in a solid infinite in three dimensions, finite stresses are not 
transmitted undiminished from infinity, as in a rod or lamina. The solutions, in 
fact, will lead to stresses that become zero at infinity. This has been shown by 
Boussinesq Ajjplications des Potentiels a I’Etude de lEquilibre et du Mouvement 
des Solides Elastiques,’ Paris, Gauthier-Villaes, 1885), who has interpreted Laaie 
and Clapeyron’s results, and obtained by a new method simple expressions for the 
stresses in an infinite solid, due to arbitrary surface forces applied to a bounding 
plane. The same results have lieen obtained by Professor Cerruti (“ Bicerche 
intorno all’ Equilibrio de Corpi Elastic! Isotropi,” ‘ Reale Accademia dei Lincei,’ 
vol. 13, 1881-2) in a different way. 
Boussinesq, on p. 280, suggests a possible ajiplication of his method to the case 
of two parallel planes, but he makes no attemjit to follow it up. 
In two papers in the ‘ Comptes Bendus’ (vol. 94, pp. 1510-1516, and vol. 95, 
pp. 5-11) he has considered the case when the problem of the infinite plane may 
be treated as two-dimensional, and there he has tried to extend his method to two 
parallel planes, but had to fall back upon an assumption mathematically unjustifiable. 
§ 44. Recapitulation of Results and Conclusion. 
Looking back upon the results obtained, we see that the general solution given 
has enabled us to deal with all the most important statical problems connected Avith 
the elastic equilibrium of a long beam, of finite height, in so far as the approximation 
involved in treating them two-dimensionally is valid ; and it will be A^alid, if the 
horizontal dimension of the cross-section be either Amry small or A^ery great. 
Incidentally the question of the effect of concentrated loads, whether in the form 
