138 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
cosh kh sinh Kh 
v= (= eoshae + Sak cosh xz cosh xz’ = | ne sinh xzsinh x2’ )J eR : 
Next we replace the term 1/r in V by the equivalent integral 
| eFRS KRdk. 
0 
Hence 
h xh ; , Sinhixiie : ' Cas 
V= i | (#sinh ae? 4 COST cosh xz cosh xe! ~ “T° sinh xz sinh Ka )TyR = = hak, 
the upper or lower sign being taken in the ambiguous term according as z is > or < 7%. 
When R>0, this integral can be separated into the two 
ic J rR F sinh z—2 + = 2 cosh xz cosh xz’ — e = sinh xz sinh x2’ — =) 
KIL 
° 8h 
ie i (JoeB -¢ 1 
R 
The value of the latter integral we have found to be — = log 5, ° 
The former integral is of the form | ey ok RE(«)de, where F(«) is an odd function of «x, 
vanishing for«=0. It may be expressed as a complex integral 
1 
= [GeRE(e)a, 
the path being from west to east along the whole of the real axis in the « plane, for 
G(«R) — Gy(xeR) = iT xR. 
Now, from the original form of V,, and the integral forms of 1/7, it is obvious that 
F(«) vanishes at infinity in the eastern half of the « plane; being odd in « it must 
vanish likewise in the western half. 
Hence by Cauchy’s Theorem, the integral = i G «RF («)d« is equal to twice the sum 
of the residues of the function G)«RF(«) at its poles in the upper half of the « plane, 
and 
R 
V= = 7,108 a6 
nr . inrR 
2 cos a = e087, Go h 
oe 1 \7z.. 1 \7z’ 1 \77rR 
5 2 si n(n + 5 \esin (n + a) i Go(m oo oe : 
(j) The solution indicates (i) a main current in two dimensions, defined by the 
; 1 R ais : : : : : : 
potential — 7 log,,, and (ii) an infinite series of local currents in three dimensions, 
