268 
DR. G. B. JEFFERY ON PLANE STRESS AND 
§ 2. The Co-ordinates. 
Let us take curvilinear co-ordinates defined by the conjugate functions 
, -a i x + %(y + a) 
a \-tft = log- Tj -C 
x-H (?/— a) 
(1) 
where x , y are Cartesian co-ordinates and a is a positive real length. Solving for 
x, y, we have 
a sin ft a sinh a 
x - ;--— y - —,-—77.(2) 
cosh a — cos ft ' ' cosh a — cos ft 
Elements of arc measured along the normals to the curves a, ft = constant are 
respectively Sa/h, Sft/h, where 
2 
1 
/r \ca 
oy 2 
\C a 
from which we have 
h = (cosh a — cos ft)/a. 
(3) 
The general scheme of co-ordinates is shown 
in fig. 1. If 0 1( 0 2 are the points 0 , —a and 
0, a respectively and P any point in the plane, 
and if the radii from Ch, (X to P are of lengths 
r 2 and are inclined at angles 6 lt 0 2 to the 
axis of x, then a = lo grjr 2 and ft = — 
The curves a = constant are a set of co-axial 
circles having O l5 0 2 for limiting points. The 
circles corresponding to positive values of a lie 
above the rr-axis and those corresponding to negative values below, while the avaxis 
itself, which is the common radical axis, is given by a = 0. The curves ft — constant 
are circles, or rather arcs of circles passing through O,, 0 2 and cutting the first set of 
circles orthogonally. On the right-hand side of the y -axis ft is positive and on the 
left-hand side negative, while on the //-axis 6=0, except on the segment OjO^, 
where ft = ± tt . At infinity a = 0. ft = 0, and at O,. 0 2 we have a = — 00 and + 00 
respectively. 
We have thus a set of co-ordinates adapted for the consideration of two-dimensional 
problems in which the region considered is— 
(1) A finite region bounded internally by a circle and externally by a larger and 
non-concentric circle. 
(2) A semi-infinite region bounded externally by a straight line and containing a 
circular hole. 
(3) An infinite region containing two circular holes of any radii and centre 
distance. 
