174 Transactions of the Royal Society of South Africa. 
where a is the radius of the ring. 
The velocity due to an infinite plate- source will therefore be 
^ a^da 
(^2 _^ a2)t 
where Cj is the strength of the distribution. 
We therefore have : 
For Z > 0, V ■= ■\- 47r2c 
_ 
1 - 
For z < 0, V = — 47r2c^ = — 
where (p'^ is the velocity-potential of the motion. 
Further, ^>'o 
Hence for z > o, *l>'^— — 47r2cj^| 
2 < 0, = 4- 4>7r^C-^z\ 
(27) 
2. Circular Plate Sink. 
From 22 we have that the velocity-potential of a ring sink of radius a 
is given by 
^-,(«)l;(2-t'^)K-2Ej, 
where the modulus k is given by 
k^ 4ar 
^2 +(« -!- ry^ 
The velocity-potential due to a distribution of sinks over a circular 
plate of radius «^ will therefore be given by 
0" -= _ 47rci y ~ . )^ I (2 - yfc2) K - 2E I da. 
o 
Now, the modulus, k, is less than one for all values of z and r, excepting 
^ = o, ?• = a, where k^ = 1. 
We may therefore write : 
1.3\^, . , /I. 3. 5' 
K 
E 
These expansions are valid at all points excepting the ring of points 
: a in the plane z =: o. 
