274 
DR. G. B. JEFFERY ON PLANE STRESS AND 
and the corresponding terms in hQ 
— ^ - hQ = (E cos (3 + F sin (3 + G cosh a 4- H sinh a) a 
A + 2m 
-(Bo cosh a+D 0 sinh a + Dj cos /3 + D', sin (3) (3. 
From (19) and (20) we may now find the corresponding displacements u, v. Each 
of these is found to contain a multiple of the many-valued co-ordinate (3. Equating 
the coefficients of these terms to zero we have the following relations 
E + G = 0, Bq + Dj = 0,1 
mF— (x + 2 ju) I) 0 = 0, //H + (\ + 2 /i) = 0. J 
We shall now show that these early terms correspond to the resultant of the forces 
and couple applied over the boundaries. For this purpose we shall require the following 
elementary forms of the stress-function :—• 
(1) For an isolated force X applied at the origin in the direction of the x-axis 
x = -{2ir)- l X{y6-vx log r) 
where r, 0 as usual denote polar co-ordinates and v = /u/(\ + 2/x), 
(2) For an isolated force Y applied at the origin in the direction of the ?/-axis 
* 
X =(27r)~ 1 Y (x6 + vylog r). 
(3) For a point couple of moment L applied at the origin in a positive sense 
X = -(27r)" 1 La 
(4) For a centre of pressure radiating uniformly from the origin 
x = log r, 
Inserting the relations (23) necessary to ensure single-valued displacements our 
early terms become 
or 
h x — G (cosh a —cos (3) (3 + (3 0 (cosh a—cos (3) a 
+ F (f3 sin (3 + va sinh a) + H (/3 sinh a — va sin (3) 
X = «G/3 + aB„a + F (x/3+vya) + H {y(3—vxa) .(24) 
Now aG/3 = aG(6 1 — 9.Q and hence this term represents a couple of moment 2-rraG 
applied at a = co and an equal and opposite couple applied at a = — o°. The term 
a/3 0 a represents two equal and opposite centres of radial pressure at these same points. 
