Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

280 DR HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC,
GREEN'S FUNCTION FOR THE CIRCULAR DISC.
3. In order to obtain Green's function for an indefinitely thin circular disc, which
we take to coincide with the fundamental circle of our system of coordinates, we shall
apply the idea originated and developed by Sommerfeld*, of extending the method of
images by considering two copies of three-dimensional space to be superimposed and
to be related to one another in a manner analogous to the relation between the sheets
of a Riemann's surface. In our case we must suppose the passage from one space to
the other to be made by a point which passes through the disc; the first space is
that already considered, in which 0 lies between - r and rr; for the second space we
shall suppose that 0 lies between 7r and 3wr, thus as a point P starting from a point
in the first space passes from the positive side through the disc, it passes from the
first space into the second space, the value of 0 increasing continuously through the value
wr, and becoming greater than 7r in the second space. In order that a point P starting
from a position PO(po, 00, bo), say on the positive side of the disc, may after passing
through the disc get back to the original position P0, it will be necessary for it to
pass twice through the disc; the first time of passage the point passes from the first
space into the second space, and at the second passage it comes back into the first
space. Corresponding to the point p,, 00, <0 where 00 is between -7r and 7r, is the
point (po, o + 27r, <Po) in the second space, whereas the point (po, + 47r, 0o) is regarded
as identical with the point (po, 0o, <0). The section of our double space by a plane
which cuts the rim of the disc is a double-sheeted Riemann's surface, with the line of
section as the line of passage from one sheet into the other. Let p,, 0o, 0o, be the
coordinates of a point P in the first space, on the positive side of the disc, thus
O < 00< wr; taking the expression for the reciprocal of the distance of a point Q (p, 0, 0)
from P, given in the last article, we have, since
sinhu sinh 1 sinh 1
sinh u _1 _ 2 __ 1 __ 2 _sh
coshu-cos (0- 00) 2 eoshI 2- s+ eh + 2 +t2 i
c2(eosh u - coso ( - 0 o) co2 sh 1 u cos 0- 0o)
sinh u
i i.. n\i /, ~~i rOI) 12
pQ = 2 V/2wra (coh P - cos 0)* (,coshl Pi - cos 0j - -osh - cosh -cosh -sinh -u du
2Q 2 - cos 1 (0 - 0,)
sinhU
+ 1- (cosh - cos 0)i (cosh P - cos c du
2 /27ra a /eosh u - cosh a eosh u - os - 27r).
we 1~ ~ / aastto
we thus see that 1/PQ is expressed as the sum of two functions, the first of which
involves the coordinates po, 0o, <po of P, and the second is the same function of the
* See his paper "Ueber verzweigte Potentiale im Raume," Proc. Lond. Math. Soc. Vol. xxvIII.