BEAM OF EECTANGULAE CEOSS-SECTION UNDEE ANY SYSTEM OF LOAD. 153 
More recently the same problem has been attacked by M. C. Eibiere in a thesis 
(‘‘Sur divers Cas de la Flexion des Prismes Rectangles,” Bordeaux, 1889 ; see also 
‘ Comptes Rendus,’ vol. 126, pp. 402-404 and 1190-1192) in which he gives a solution 
in a series of circular and hyperbolic functions. He takes his plate of finite dimen¬ 
sions and built-in (encastree) at the edge. By this term he understands that the 
edge is constrained to remain plane and vertical, and is subject to no shearing-stress. 
For other terminal conditions the solution, as M. Ribiere states himself, is insufiicient. 
I find that, if the edges of the plate be removed to infinity, his solutions degenerate 
into Lame and Clapeyrox’s integrals, of which they therefore give the true 
meaning. 
M. Ribiere, in the same thesis, has also investigated the two-dimensional case, 
which has been treated of in the present paper. ^ I am indebted to M. Ribiere for 
very kindly communicating to me his thesis, with which I became acquainted after 
my woiF had been completed. His solutions are of the form (26) (27) (28), and he 
determines his coefficients, as far as I can see, by the method used here, but does 
not transform his expressions further. Like Lame and ChapeyroxX, he restricts his 
applied surface stresses to be normal and investigates only two sjcecial cases. 
M. Ribiere takes, as I have done, m = mr/a. This, by the v^ay, is not absolutely 
necessaiy. Another set of solutions might be obtained by taking m = (2u +1) 7 r/ 2 a. 
When a is made very large, as is the case in every one of the problems treated here, 
either set of solutions will lead to the same final form, provided the total terminal 
conditions are attended to. M. Ribiere, on the contrary, in order to be able to 
evaluate his series, which become far more manageable when 6/a is large, treats 
chiefly of cases of thick beams of very short span. Now in this case it is no longer 
permissible to consider merely the total conditions over the ends x= ±a, and to 
treat the actual distribution over these ends as unimportant. M. PtiBiERE gets over 
this difficulty by supposing his beam to be encastre, as defined above. The same 
mathematical condition of fixing is assumed by Professor Pochhammer (‘ Crelle’s 
Journal,’ vol. 81) when treating in a similar fashion of the beam of circular cross 
section. 
It seems doubtful whether anything of this kind does really occur at an actual 
built-m end of a beam. Certainly Pochhammer and Ribiere’s conditions do not 
agree with the view taken by be Saint-Venant, who, in his calculation of the 
deflection for a cantilever, has assumed that the elastic line is not horizontal at the 
built-m end. In this case, however. Love has pointed out that the elastic line may 
have any small slope at the budt-in end, provided we superimpose a suitable rigid 
body displacement. But both he and be Saint-Venant agree to make the ^d 
* Since writing the above, I find that Professor Lamb (‘Proc. Loud. Math. Soc.,’ vol. 21, p. 70, paper 
read December, 1889) has worked out the same problem in the form of a series of circular and hyperbolic 
functions, but he has left his results in this form, without interpreting them further, and I cannot discover 
that he has considered end-conditions. 
VOL. CCI.-A. 
X 
