The Torsion Problem for Bodies of Revolution. 
171 
Section 11. 
Ring Do^ihlet. 
Let a circular plate perpendicular to the .2; axis, its plane passing 
through the origin, be uniformly and continuously distributed with doublets 
having their axes perpendicular to the plane of the plate. If is the 
strength of the distribution, then the velocity -potential of an element 
^x.^ Sx^ dx'r^, whose co-ordinates are (0, x'o, x'.^, x'^, x'-) at a point {2, Xo, 0, 
0,0) will be given from equation 18 by 
c Kn cos 0 
C(p- = — -i — y - . SXc^ dx.. Sx' Sx'-, 
where 9 is the angle between the axes of the doublets and the line joining 
the points (0, x'^, x'.^, x' ^, x'-) and {2, x^, 0, 0, 0), and p the distance between 
these points. 
Now p2 _|_ (/j.^ _ ^'^^2 _j_ x'.^^ -f ^ x'J^, and the direction-cosines of 
this line are given by 
Xj^ ^3 ^4 ^,5 1 
2 1 Xn X •! — X ■t — X A X ^ o 
The direction-cosines of the axes of the doublets are 1, 0, 0, 0, 0. 
2 
9 ' 
Hence cos 0 = — 
U'2 ^-^^'3 ^^'4 ^'^'s 
Hence the velocity-potential due to a distribution over a circular plate 
of radius will be given by 
^X'.y dX'.. SX'_^ SX - 
_ i ^ r r r r ^ ^^'-^ ^^'5 ! 
~ ^"177./ J J J ( {^+7^7- a:^^"^^ 
where the integration is extended over the finite region 
x\;} + x'.^ + X + ^'h'^^^H^ 
and in which region the function integrated has no singularities provided 
2 0 in the region x_^^ay 
From the above we see, then, that the velocity-potential of a ring 
doublet is given by 
where ^'g is the velocity-potential due to a ring source. 
We therefore have for the velocity-potential of a ring doul)let of radius a 
hz ( 1 -f h"' 
