166 Transactions of the Boyal Society of South Africa. 
Or if we go over to the limit for ^a = o, and take L ^i^^ ^ where c is 
Sa 0 
finite, then we get for the velocity-potential of a ring source of radius a 
^ J {z^ a- - 2ar cos a.U 
Putting TT - a. = 2x, the result reduces to 
I sin^ 2x . dx 
Mc ri sin^ 2x . 
where — — ^nd Av = i/l _ A^^iTi^A/- 
Let us now consider the integral 
We have the identity : 
d /sin X cos %\ _ F sin' x c^^' X _|_ cos' x sin'^ x 
_ P sin2 2x 1 2 sin' 
~ 4 ■ A^X Ax Ax " 
Integrating between the limits o and x» we get : 
sin^ 2x . fZx _ 4 sin x cos 8 sin- ^ • (h _ ^ n - dx 
J A '^X ~ ^'o A X A:" ./ Ax ^' J 
But 
Ax 
O " 0 
./Ay Tr %J £i.x ^ ¥• ' 
where F (x) and E (x) are Legendre's elliptic integrals of the first and 
second species respectively. 
We therefore have 
J Ah AX "^'^^ j 
For the amplitude x = ^, we therefore have 
I, = /-5f..x = >:(2-.^)K_2E: . ■ - (20) 
o 
where K and E are the complete elliptic integrals of the first and second 
species respectively. 
