PLANE STRAIN IN BIPOLAR CO-ORDINATES. 
267 
two concentric circles, or an infinite plate containing a circular hole under any 
given tractions applied over its boundaries. In his lectures at University 
College, London, in 1912, Prof. Filon gave the complete solution of this problem 
determining the stresses and displacements when the stresses on the boundaries 
are expanded in Fourier series, and I am not aware that this solution has ever 
been published. An outline of the solution in elliptic co-ordinates is given in Love’s 
‘ Elasticity.’* 
In this paper the complete solution is given for bipolar co-ordinates, for which the 
co-ordinate curves are co-axial circles. This solution enables us to treat the problems 
of an infinite plate containing two circular holes, a semi-infinite plate bounded by a 
straight edge and containing one circular hole, and a circular disc with an eccentric 
circular hole. 
In the second Section the equations are expressed in bipolar co-ordinates and 
formulae are established for the displacements in terms of the stress-function. 
In the third Section the stress-function is obtained in a convenient form and the 
terms giving rise to many valued displacements are separated out. 
The fourth Section is devoted to the determination of the coefficients in the stress- 
function when the tractions over the boundaries are given in Fourier series, and to an 
examination of the convergence of the resulting series. From the results established 
in this section it appears that the solution is complete, for the stress-function can 
always be uniquely determined when the tractions are given, provided that the 
applied forces taken as a whole are in equilibrium. 
The remaining sections are occupied with the examination of some of the simpler 
applications of the theory. Section 5 gives the solution for a circular disc with an 
eccentric hole (or a cylinder with eccentric bore) when the two boundaries are under 
different hydrostatic pressures. It is found that the solution of this problem can be 
expressed in finite terms. An important particular case of this problem is discussed 
in Section 6, namely, a semi-infinite plate with a straight, unstressed boundary and a 
circular hole under a uniform normal pressure. This will give the stresses near 
a rivet hole while the hot plastic rivet is being forced home under pressure. 
This solution is interesting from another point- of view, for if the ratio of the 
radius of the hole to its distance from the edge is suitably adjusted, the point of 
greatest tension will be on the straight edge while the point of greatest stress 
difference is on the circular boundary. It thus suggests a crucial test for the 
rival theories of rupture,—the greatest tension theory and the greatest stress- 
difference theory. 
Section 7 deals with a semi-infinite plate with an unstressed circular hole 
under tension parallel to its straight edge. The solutions are in the form of infinite 
series, but the more important aspects of the problem are illustrated by numerical 
tables. 
* 2nd edition, p. 259. 
2 p 2 
