176 Trmisactions of the Royal Society of South Africa. 
={p-2n+^) r(2p-27i + 3) + (^^ + r^) Ip-i, 
which is obtained by differentiating [z'^ + {a -f r)^] with respect to a, 
and integrating, and where 
"1 
ap . da 
^ [^2 + (a + r)2f "2 
The integral may again be evaluated by applying the transformation 
a -{-r= z tan I. 
We get 1 
lo = /2(«-]) /'(I - at 
tan~-^-^ 
where t = sin |. 
If now 0y = (p'^ + 0^8, 
then will be the velocity-potential due to a uniform distribution of 
sources over an infinite plate with a circular hole in it. The stream-lines 
■of such a distribution will be approximately such as shown in Plate XYIII, 
If we superpose upon this motion a uniform stream of velocity Vq, whose 
velocity-potential <po is from 14 given by 
06 = — VoZ, 
then the stream -lines of the motion, whose velocity-potential is given by 
08 = 00 + 08 
will approximately be as shown in Plate XVIII, fig. 2. 
From this, then, we get a body such as shown in Plate XYIII, fig. 3. 
Within such bodies, then, the stress-components will be given by 
dr 
and the displacement ue by 
02 = r . -P- 
the constant c determining the surface of zero displacement. 
At 2^ = — X, V-<x: — Vo — 47r2cj. 
At ^ = + oc, 1;+ oc = + 47r2cj. 
The moment of the terminal couples will therefore be given by 
M = - J J^ 7-2 (S) ^dr .dd^-fjr''^ (ez), = -o.dr . de , 
