160 Transactions of the Royal Society of South Africa. 
Section 8. 
Uniform Stream and Two Doublets. 
Let us noNv investigate the motion due to a uniform stream parallel to 
the ;2-axis superposed upon that due to two doublets of equal strengths at 
the points z = -\- a and z — — a. 
From 14 and 18 we have for the velocity-potential of such a motion 
'k{z -\- a) Jc (z — a) 
0, = -Voz- ^^^2_|_^2}t ~ {(,_a,)2 + r2}t 
and from 15 and 19 we have for the stream-function 
Again, if we put Vo = 4, h— 1, we have 
1 1 ""I 
J/, = — r' -f : + , . 
' a)" + ^-^1^ {(^ - ^0' + r^}^ 
This family of curves may be obtained in the following way : 
Firstly, the family of curves given by 
sin'^f' 
is traced on transparent linen paper (see Plate A). This is now laid on 
Plate VIII, the two ;>;-axes coinciding, and the doublet of Plate A being distant 
2a from the origin of Plate VIII. On top of this another strip of linen paper 
is placed, and on this paper the curves, got by combining the curves C of 
Plate VIII and the curves of Plate A, are traced. In this way, then, we arrive 
at the stream-lines due to the uniform stream and the two doublets. 
The stream-lines = o are given by r = o and 
1 1 _ 
{(7+'^)2Tr^ ^ {(^ - +^ " 
This is a closed curve with symmetry about the z and r axes, and if this is 
taken as one of the boundary lines of the body of revolution in a meridian 
plane, then again we have the case of a circular cylinder with a " flaw " on 
its axis, the dimensions of the " flaw " being small compared to the radius 
of the cylinder. By the above method this curve has been obtained for 
a — ~ (see Plate B). From this we get a body such as shown in Plate XIII. 
In such bodies the stress-components will be given by 
rO — r 
^ z -\- a z — a I 
9z r=i r . — = — Vor -f- hr •! — . — . - 
