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IX. Plane Stress and Plane Strain in Bipolar Co-ordinates. 
By G. B. .Jeffery, M.A., D.Sc.. Fell ow of University College , London. 
Communicated by Prof. L. N. G. Eilon, F.R.S. 
Received May 15,—Read June 24, 1920. 
§ 1. Introduction. 
The problem of the equilibrium of an elastic solid under given applied forces is one 
of great difficulty and one which has attracted the attention of most of the great 
applied Mathematicians since the time of Euler. Unlike the kindred problems of 
hydrodynamics and electrostatics, it seems to be a branch of mathematical physics in 
which knowledge comes by the patient accumulation of special solutions rather than 
by tl le establishment of great general propositions. Nevertheless, the many and 
varied applications of this subject to practical affairs make it very desirable that these 
special solutions should be investigated, not only because of their intrinsic importance 
but also for the light which they often throw on the general problem. One of the 
most powerful methods of the mathematical physicist in the face of recalcitrant 
differential equations is to simplify his problem by reducing it to two dimensions. 
This simplification can only imperfectly be reproduced in the Nature of our three- 
dimensional world, but, in default of more general methods, it provides an invaluable 
weapon. 
It was shown by Airy* that in the two-dimensional case the stresses may be 
derived by partial differentiations from a single stress function, and it was shown 
laterf that, in the absence of body forces, this stress function satisfies the linear 
partial differential equation of the fourth order V 4 ^ = 0, where V 4 = V 2 . V 2 , and V 2 
is the two-dimensional Laplacian c 2 /dx 2 + d 2 /dy 2 . 
It might have been expected that these results would have opened the way for 
a theory of two-dimensional elasticity of the same generality as the two-dimensional 
potential theory. This has not, however, been the case. This is due in part to the 
greater analytical difficulties which attend the discussion of the two-dimensional 
* ‘ Brit. Assoc. Rep.,’ 1862, p. 82. 
t W. J. Ibbetson, ‘ Proc. Lond. Math. Soc.,’ vol. xvii., 1886, p. 296. For a history of this part of the 
subject see Love’s ‘ Elasticity,’ 2nd edition, p. 17. 
vol. ccxxi.— a 590. 2 p 
[Published November 8, 1920. 
