228 THE EQUILIBRIUM OF AN ISOTROPIC ELASTIC PLATE. ; 
ADDITION TO PAPER BY J. DouGALL ON— 
“An ANALYTICAL THEORY OF THE EQUILIBRIUM OF AN IsoTrrRoPic ELAstTic PLATE.” 
(Note added May 20, 1904.) 
The kindness of one of the referees enables me to supply the following references 
to recent work bearing on the subject of the paper. 
(a) J. H. MicHEtt, in a paper “On the direct determination of Stress in an Elastic — 
Solid, with application to the Theory of Plates,” Proc. Lond. Math. Soc., vol. 31, 
1899, shows how the stresses might be found without previous determination of the 
displacements. In the case of the stress zz or R, he finds that y*R is a given function 
in the body of the plate, while R and dR/dz are given on the faces. If we neglect the 
conditions at the edge, which have practically no influence on the result, a value of R ~ 
satisfying these conditions can be found, in terms of Fourier integrals for instance. Mr 
MrcHELL does not determine R—this has been done in the present paper—but proceeds — 
to deduce the forms of the remaining stresses, and the differential equation for the 
normal displacement of a point on the mid plane. One special case of normal force is 
worked out to a first approximation, and Lagrange’s equation for this case deduced. , 
For the conditions at the edge, reference is made to the ordinary Thomson- 
Boussinesq theory, which uses the principle of equipollent loads. 
((b) L. N. G. Fiton, “On an approximate solution for the bending of a Beam of 
rectangular cross section under any system of Load, with special reference to points of con- 
centrated or discontinuous Loading,” Phil. Trans. R. Soc. Lond. (Sec. A), vol. 201 (1903). 
Dr Fitoyn’s solution apples to a beam in which the ratios of breadth to depth, and 
of depth to length, are both small. The axis of z being taken in the direction of the 
breadth, the stress 2 is taken as negligible, and equations are deduced for the mean 
values, across the breadth, of the displacements u, v. ‘These equations are the same as — 
equations (90), page 182 of the present paper, with the body force null. In order to see” 
the reason of this from our standpoint, we may notice that the assumption that % 
vanishes eliminates all the solutions of what we have called the dilatational transitory 
type, and that taking the mean of the displacements eliminates all the flexural solutions, 
as well as the rotational transitory solutions. A 
As regards the conditions at the ends, the beam is treated as a long rod. 
It may be of interest to remark that the results of § 43 above furnish the data for 
a more approximate treatment of the problem on the lines followed by Dr Finon. 
(c) A note appended to a paper by Professor Lams in Proc. Lond. Math. Soe, 
vol. 34 (1902), pp. 288, 284, contains a solution of a special case of the problem of 
face traction. j 
(d) In connection with existence theorems relating to the elastic equations, reference 
should be made to the work of Italian elasticians, as Somictiana, LAURICELLA, and 
TEDONE. 
PRESENTED 
9 FEB. [Yub 
