PLANE STRAIN IN BIPOLAR CO-ORDINATES. 
275 
We also have 
K(y/3-vxa) = H {{ij + a) 6 x -vx\og r x } — H{(t/-a) 0 2 -vx log r 2 } 
—aH (@j + 0 2 ). 
This corresponds to a force 2-7rH applied at a = + 00 parallel to the a:-axis and an 
equal and opposite force applied at a = — co 
(thus forming a couple of moment 47raH) and 
point couples each of moment 2-TraH applied 
at these same points (see fig. 2). 
Finally a. - o 
F x/3+vya.) — F {xO l + v{y + a)\ogr^\ 
-F {x0 2 +v(y-ct) log r 2 } 
—avF log r 
This corresponds to forces each equal to 2ttF, 
acting at the points a = ± 00 and each directed 
towards the origin, together with two equal like centres of uniform pressure at the 
same points. This brings to light a new solution corresponding to the last term. 
Expressed in our co-ordinates we have 
log r x r 2 — 2 log (2a) —2 log (cosh a —cos /3), 
and the corresponding form of hx is, apart from constants, 
hx = (cosh a —cos (3) log (cosh a —cos /3). 
It is easily seen that this can be expanded in a Fourier series which is included in 
our general expression for hx, but that the expansion is different on opposite sides of 
the line a = 0. For this reason we shall find it convenient to include a term of this 
form whenever the region under consideration includes parts above and below the 
axis of x, i.e., when it is bounded by two circles neither of which encloses the other. 
It will be noted that, taken together, the early terms allow for the most general 
resultant forces acting over the two circular boundaries enclosing the two points 
a=+oo, a — — co, subject to the condition that the forces acting over the two 
boundaries considered together form a system in equilibrium. If it is desired to 
investigate problems for which this condition is not satisfied we can readily obtain the 
necessary additional solutions. They will be 
X = iy + a) 6,-vx log r/ 
X = x6 1 + v(y + a.) log r, 
X = °i 
(25) 
2 Q 2 
