306 REPORTS ON THE STATE OF SCIENCE, ETC. 
If A, u are the elastic constants of Lamé, we have, in the first case, the stresses 
ne, xy; yy given in terms of the displacements u, v by the equations . 
te =a 5" +57) 9 ou 
a By bu | 
— on br 5 | 
YyrHr ui : 2 ae . 5 a 7 4 
YY eae, + Mey (4) 
oS = bu ov 
Ht (sy =) ! 
In the second case P, Q,S are given in terms of the mean displacements U, V by 
precisely similar equations, A being replaced by A’, where A’=2Au/(A+2u). The 
two problems are thus analytically the same, and in what follows we shall confine 
ourselves to the second one, which is known as that of generalised plane stress. 
2. The conditions at a boundary, where is the direction of the outwards normal 
whose direction cosines are J, m, are given by 
X=/P' + mS 
Y=15 + mQ } ; : ‘ : C - 5) 
X and Y being the mean stresses across the boundary parallel to the axes, 
Equations (5) can be written in the form 
-_ 5 /5E - & bh 
X= —( — tose oe (Se \) ee 5 : ¢ - j 
dbs ) as( ie) (6) 
ds being an element of the boundary. 
Since the differential equation (3) satisfied by E_ and the boundary conditions (6) 
for E do not involve the elastic constants, it would at first sight appear that the 
solution for E, for any given set of boundary stresses, cannot involve the elastic 
constants and that the stress-distribution in a plate, under given applied boundary 
stresses in the plane of the plate, is independent of the material of the plate, 
provided only it be elastic. 
This proposition, if true, is of great practical importance, because it shows that if 
we can find the stress distribution in a plate by any method whatever. for example 
by the exploration of a plate of xylonite by means of polarised light, the results, so 
far as the stresses are concerned, can be transferred immediately to a plate of any 
other material, such as steel, provided the applied stress-system is the same. 
It was first shown by J. H. Michell (1) that the theorem in question does not 
hold when the area of the plate is multiply connected, that is, when it contains one 
or more holes, so that closed circuits can be drawn inside the plate, which cannot be 
made to shrink into a point by continuous deformation without passing outside the 
material of the plate. 
The problem was discussed by A. 'Timpe (2) in connection with a ring boundary 
bounded by concentric circles, and Timpe was the first to point out that the failure 
of the theorem was due to the existence of types of strain in such a multiply-- 
connected plate, created by removing a thin strip or wedge of material, thus cutting 
a thin channel or gap, with straight edges, from the hole to the outer boundary and 
then closing up the gap and cementing its faces together. We then obtain a stressed 
ring, although no forces are applied to the circular boundaries. 
More recently Prof. Vito Volterra (3) has given a general theory of such types of 
strain, to which he has given the name of distorsions, and for which Prof. A. BE. H. 
Love (4) has proposed the English term dislocations. Prof. Volterra’s account is not 
restricted to two dimensions, and he has incorporated in it various scattered results . 
obtained by Weingarten, Cesaro, and Tedone. He does not, however, discuss the _ 
question whether the solution is independent of the elastic constants, as this question 
does not really arise in the three-dimensional case. So far as I have been able to 
find out, Michell’s has been the latest statement on this particular subject. 
3. Thisresult of Michell’s necessarily throws doubt upon the general applicability 
of investigations with xvlonite models to ordinary engineering structures, since the 
ratio of the elastic constants is different in xylonite and in the usual engineering 
materials. M. Mesnager (5), in answering a similar objection to his use of glass models, 
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