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.

282 DR HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC,
4. To evaluate the definite integral in the expression for W write cosh u =S
I 1(i-0),th
cosh a = -, cos (0 - 00)=, then
2 2
sinh 2 u
l- os sl 2 --- --- - du='/-2 dx
/c - cosa scosh - cosh (O o ) - 2 ( - rT)
=/2 /- 72 5-S, + Sm-n-'
/ v-' ( '2 ( -)
where the inverse circular function has its numerically least value; we thus obtain the
expression
1 (cosh p - cos 0) (cosh p, - cos 6) r7. ( 1 1 1
W v = c-oo [j- + sin- cos _ (0- o0) sech a1,
7raV\/ 2 {cosh a - cos(- o)2 L(2- )sech + n
which may also be written in the form
W=Q + sin-l cos (0-00) sech..................... (1).
This expression TW has the following properties:-it is, together with its differential
coefficients, finite and continuous for all values of p, 0, 6 in the double space, except
at the point P in the first space, and it satisfies Laplace's equation; when Q coincides
with P, the inverse circular function approaches, and the function becomes infinite
as 1/PQ; when however Q approaches the point in the second space which corresponds
to P, the inverse circular function approaches -, and the function does not become
2'
infinite. The expression (1) is then the elementary potential function which plays the
same part in our double space as the ordinary elementary potential function 1/PQ does
in ordinary space.
5. In order to find a potential function which shall vanish over the surface of
the disc, and shall throughout the first space be everywhere finite and continuous
except at a point P (po, 0o, 0o) in the first space on the positive side of the disc
(0< 0o <7r), we take the function W(po, 0o, bo)-W(po, 2wr-00, bo) which is the
potential for the double space due to the point P and its image P'(po, 27r-0o, bo),
which is situated in the second space at the optical image of P in the disc. This
function is equal to
1 (cosh p - cos O) (cosh p, - cos!0 (- +0) sech I af |
7rc /2 {cosh a - cos (0 - 0o)} L2 ( 2 2 )
1 (cosh p - cs 0)t (cosh p0 + cos 0) 0 r. + I 41 )
7ra42 {cosh a + cos(0+ 00)} L 2 / 2