266 
DR, G. B. JEFFERY ON PLANE STRESS AND 
solutions of V 4 \ = 0 as compared with V 2 y = 0 . The analogues of many of the 
important properties of the simpler equation have yet to be discovered if they exist at 
all. Some progress has been made, and in this connection we may mention the work 
of J. H. Mighell* who established a general theory of inversion which, with some 
important differences, follows the potential theory fairly closely. 
No doubt the analytical difficulties have been the chief obstacle to progress, but 
perhaps the theory has not in recent years received attention which it would have 
received but for a certain physical difficulty. A truly two-dimensional elastic system 
is not so easy of realisation as might seem to be the case at first sight. If the stresses 
are everywhere parallel to the xy plane and independent of 2 there will in general be 
a varying displacement parallel to 2. If the displacements are everywhere parallel to 
the xy plane and independent of 2 this can only be secured by the application of 
a stress 22 which varies from point to point and is perpendicular to the xy plane. This 
difficulty was in a large measure removed by a theorem established by Filon, which 
has been called the theorem of generalised plane stress, t It states that if the average 
value of the stress 22 be taken throughout the thickness of a plate parallel to the xy 
plane, then the ordinary two-dimensional theory will give accurately the average stresses 
through the thickness of the plate if the elastic constants of the material are modified. 
If A, m denote the true elastic constants, A must be replaced by W = 2 \/u/(\ + 2 fx) while 
[x remains the same as before. This theorem attains an even greater importance when 
considered in the light of Michell’s theorem,^ that if a plate bounded by any 
number of bounding curves is in equilibrium under forces in its plane applied over the 
boundaries, then, provided the forces applied over each boundary taken separately 
are in equilibrium, the stresses are everywhere independent of the elastic constants. 
The hypothesis that the average value of 22 vanishes throughout the plate, 
while certainly not accurately true in the majority of cases, will probably give 
a very close approximation in the case of a thin plate where parallel faces are 
unstressed. 
In the light of this generalisation it is of considerable importance that the two- 
dimensional problem should be worked out more thoroughly. The two-dimensional 
solutions of V 4 ^ = 0 have been investigated in several systems of curvilinear co¬ 
ordinates. Owing to the special importance of the problem of the rectangular beam 
the solutions in Cartesian co-ordinates have naturally received a considerable amount 
of attention. Michele gave the general form of the stress-function in polar co¬ 
ordinates, thus opening the way for the solution of the problem of a plate bounded by 
* “The Inversion of Plane Stress,” ‘ Proc. Lend. Math. Soc.,’ 1901, vol. xxxiv., p. 134. Many of the 
results of the present paper can be obtained by an application of Michell’s methods, but it has proved 
more convenient to proceed on different lines. 
t ‘ Roy. Soc. Phil. Trans.,’ A, 1903, vol. 201, pp. 63-155. 
| ‘ Proc. Lond. Math. Soc.,’ vol. xxi., 1900, p. 100. 
