BEAM OF EECTANCtULAR CROSS-SECTION UNDER ANY SYSTEM OF LOAD. 151 
(2) The stress Q is not zero ; that is, de Saint-Venant’s assumi 3 tion, that there is 
no stress across fibres parallel to the axis of the beam, does not hold. Indeed, it was 
obvious from the beginning that it would not, seeing that tliere is a stress Q at tlie 
upper surface, by hypothesis. 
(3) The distribution of shear at each cross-section is parabolic, and is given in terms 
of the mean shear by the same formula which liolds when the shear is uniform. 
/(PY\ „ (fd — x") Q)q 
^ ~ 5ES' 
(-t) 
r7r2 / 
LIX /y = 0 
3 
4 
The curvature is therefore no longer exactly proportional to the bending moment, 
but contains an additional constant term. A similar result has been obtained by 
Piofessoi Karl Pearson and the author for beams of elliptic cross-section under 
their own weight (‘ Quarterly Journal of Mathematics,’ vol. 31, p. 90). It has 
since been shown to hold for beams of all forms of section by Mr. J. H. Michele 
(‘ Quarterly Journal of Mathematics,’ vol. 32). 
§ 43. Historical Summary : Remarks and Criticism. 
It may be of interest to give in this place a short sketch of the previous works on 
the subject, in so far as they are at present known to me. 
Lame, in his Lecons sur lElasticite (p. 156 seq.^, discusses the general problem 
of the lectangular block, with the single restriction, that the surface stresses are 
purely normal and are even functions of the co-ordinates. He fails to determine his 
constants, except in the particular case w^here the cubical dilatation throughout the 
block happens to be previously known. As this condition is never satisfied in any 
actual problem, the solution is of comparatively little use. 
De Sai.nt-Venant, in a classical memoir (‘ Memoii-es des Savants Etrangers de 
I’Academie des Sciences de Paris,’ vol. 14), has given solutions for the rectangular 
parallelepiped under torsion and flexure. These solutions correspond to tlie case of 
terminal stress-systems which are transmitted through an otherwise unstressed 
long bar. 
Numerous attempts have been made to solve the problem of the rectangular elastic 
solid by removing one or more faces to infinity, and thus simplifying the surface 
conditions. 
M. Emile Mathieu, in his treatise, ‘ Theorie de lElasticite des Corps Bolides,’ 
Pans, 1890 (see also ‘ Comptes Rendus,’ voL 90, pp. 1272-1274), has given a solution 
of the problem when it can be reduced to two dimensions. His problem is therefore 
practically the same as that of this paper, excejit that he has considered only what I 
have called case (A) on p. 66, and also, that the length a is not taken to be large and 
the distribution of stress over the faces x= a is given. The solution is, however, 
so complex in form, and the determination of the constants, by means of long and 
