152 
MR. L. X. (.J. FILOX OX AX APPROXIMATE SOLUTIOX FOR BEXDIXG A 
exceedingly troublesome seiies, so laborious, that the results defy all attemjjts at 
interpretation. 
Dr. Chree (‘ Roy. Soc. Proc.,’ vol. 44, and ‘Roy. 8oc. Archives’; also ‘Quarterly 
Journal ot Mathematics,’ vol. 22) has considered at length the solutions of the 
equations of elasticity in integral powers of x, y, z, and has applied them to the beam 
problem. Among other results he has obtained expressions for the teians independent 
of s of a foim similar to (110) of this paper. Incidentally, he verities a number of 
DE Saint-Venant’s results ; but no further application is, I think, made of the two- 
dimensional terms. 
Quite recently, Mr. J. H. Michele lias investigated the theory of long beams under 
uniform load (‘ Quarterly Journal of Mathematics,’ vol. 32, pp. 28 et seq.). The object ' 
appears to be to extend he Saint-Venant’s researches to uniformly loaded beams. 
Mr. Michell deduces several interesting results applicable to beams in general and 
to the rectangular beam in particidar, but, so far as I can see, he makes no claim to 
having obtained explicitly the complete solution in any case. 
Ihe surface conditions, however, may be thinned down still fui'ther by removing 
four faces to infinity, leaving only an infinite })late of finite thickness. The problem 
in this form has been formally solved by Lame and Clafeyron (“Sur Tequilibre 
interieur des solides honiogenes ” ; ‘ Memoires des Savants Etrangers de I’Academie 
des Sciences de Paris,’ vol. 4, pjj. 548-552). Their solution, obtained in the form of 
quadruple integrals, satisfies tlie surface stress conditions over the two infinite faces. 
The objections to this solution are two-fold. In the first place it is difficult of inter¬ 
pretation, and the integrals do not enable us to obtain a clear notion of the separate 
effects of the various forces applied to the jDlate. In the second place this solution 
takes no heed of the conditions at the other four limiting faces of the plate which, we 
should always remember, although they have been removed to a very large distance 
away, have not physically disappeared. Given total tensions, shears and couples, 
applied to the f()ur narrow faces of the plate, will j^roduce stresses that will be 
transmitted through the plate, exactly as in the case of a bent or twisted bar, and 
will produce a finite effect at points of the plate infinitely distant from the edges, 
even though the large plane surfaces should be absolutely free ffoin stress. 
In order therefore that Lame and Clafeyron’s forniulse may correspond to a 
physical reality, we must superimpose on their solution another of this transmissional 
such that the total shears and total couples due to the sum of the two solutions 
are all zero round the contour of the plate. Now the problem of the thick elastic 
rectangular plate, under given total shears and couples round its contour, but other¬ 
wise free from stress (which is the analogue foi‘ plates of the ordinary tensional and 
flexural solutions for bars), is another of the unsolved problems of the theory of 
elasticity and, until it is solved. Lame and Clafeyrhn’s solution, unless it happens 
of itself to satisfy the conditions of no total force at the edge—which vlll only be 
true ill special cases—fails. 
