164 Transactions of the Royal Society of South Africa. 
If we now take plane z = — b as being fixed, where h is great com- 
pared to a, such that quantities of the order p and higher may be 
neglected, then the displacement is given by 
ua = - ^'o(^^^)' ' , _^ 1 ^ + ^ ^ - ^ ) 
^^r \{(z + a)^ + r^y^ {(^ _ 4, ^2 
The moment of the terminal couples is where is the radius of 
the body at z = — oc . 
SectionIO. 
Bi7ig Source. 
We take the ring source such that there shall be symmetry about the 
2;-axis, the plane of the ring passing through the origin. 
In order to get the velocity-potential of such a ring source, we may 
firstly calculate the potential due to a uniform and continuous distribution 
of sources over a circular plate perpendicular to the ^-axis, and then differ- 
entiate this potential with respect to the radius, w^hich will give us the 
potential due to a ring. 
Let the co-ordinates of any point on the plate be given by (0, x'^, x' 
x\, X and that of the point P, where the potential is to be found, x.^, 
x.^, x^, x^^. 
The velocity-potential due to an element ^ ^^'^3 ^ ^ ^'^'5 
^x'o . ^^Xo . ^x\ . ^x'- 
dd)r. == c'l . ~ ' 
+ (^3 - ^'2)' + (^3 - <s)' + K - ^'4)' + (^5 - ^'5)'}*' 
where c^ is the strength of the distribution. 
For a circular plate of radius a-^ the potential will therefore be 
dx'^, dx'.^, dx\, dx\ 
= c, ffff ~ 
ix^-x^Y + - - - + (^- - x^,y}i- 
inteo-rated over the reofion 
2 
Now, by applying linear transformations we readily see that, without 
the loss of generality, we may choose our axes such that the co-ordinates 
of the point P are x^, 0, 0, 0). 
The result then reduces to : 
/*/»/»/» dx'r,, dx'.,, dx\, dx\ 
^^=''JJJJ " 
