29 T. H. Havelock. 
Putting w = « cos 0, v = « sin 9, this may be written 
1, 9) = [ dd | {F, cos («x cos 6) cos (xy sin 6) 
it, 0 
+ F, sin (kz cos @) cos (xy sin 0) + Fy cos (xx cos @) sin (xy sin 9) 
+ F, sin (kx cos 0) sin (xy sin 0)} « dic, (20) 
where 
ht, = al | F' (s, ¢) cos (Ks cos 0) cos («é sin 0) ds dé, (21) 
oT ioe) ll ae) 
with similar expressions for F,, F'3, Fy. 
If G (x, y) is another function given as a double integral in the form (20), it 
follows as in the one-dimensional case that: 
| | E («, 4) G (a, y) dx dy = 4r i. a0| (F,G,+FGo+F,GstF,G,) ede. 
—o J—0 =_ 0 
(22) 
It is assumed that the various integrals are convergent. 
For the particular case given in (18) and (19), we find that F,G, + F,G, 
reduces to a simple expression, and we obtain 
R= Lim yp | | ALD ay 
uo =odiacs GH 
pe lee 3 e-2«f dx dO 
= Lim 16pxjiPp ("| we 
eon ca o Jo (k—Kko sec2 0)2+p2 sec2 0 
1/2 
= 16rpxy!ME | SEC mmemnod acca 
0 
1 
= AmoicytM2 e~ “oF fic (kof) +(1 + =.)K (« i (23) 
0 \ 0 \"0. of i (kof 
where K,, is the Bessel Function defined by 
o 
Ka) = | e724 cosh nu du. 
“0 
Horizontal Doublets in Vertical Plane. 
4. This method allows easily an extension to any distribution of doublets. 
Consider first two horizontal doublets M and M’ at the points (h, 0, —f) and 
(h’, 0, —f’) respectively. The surface value of ¢ is now given by (16) with 
