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.

290 DR HOBSON, ON GREENS FUNCTION FOR A CIRCULAR DISC,
the value of the required function, R denoting the distance PQ. We now introduce
new coordinates r, v, b instead of p, 0, 0, these being given by
x = r2 + a2 sin. cos, y = r2 + a2 sin sin, z= rcos' ç;
to express rv in terms of p, 0, we have
x2 + y2 = (r2 a2) sin2 ( 27 + a2) sin2
Q2- a2 z2
hence cos4 7+ - a' cos2S= 2, where CQ2 =x2+ y2 + z2; hence we have
a a2
/os22 - = (CQ2 - a2)2 + 4a2z
COS2 77 \yi a + \_-_ Q 49
~2a + 2C2a2
2a2 cos 0 2a2
Now CQ2 - a2= cos - and it is easily found that /(CQ2- a2)2 + 4a2z2 = a;
cosh p - cos ' cosh p - cos 9
hence we have
1 -cos 0
cos r/ cosh p - cos '
and therefore cos v = / 1 cos 00
cosh po- cos 00
Also as P is on the plane of the disc (r=0), we have CP =a sin q, hence eP= + sin
1- sin vJ
from which we find 1 + cosh p = 2 sec2 r. Reimembering that
1 _ 1 l + cosh p Vcosh po - cos 0
R a V/2 /cosh a + cos 0o
we have AV/ 1 - cos 0O a cos 7 cos lo
cosh a + cos 0o R
and also
i I
(1 + Cshp) os 2 / + coshpp cosh 2 2
2 42, 2,
V/eh2 2- 1 i*o R' V/coshpo -cos 0 R' cos ' a /2 cos o aR cos7 cos 'o'
cosh2 a - sin2 00
then since dS = a2sin cos dd cdb, we have for the potential function at an external
point rt, qo, 0o, which has the value V(ÇY, b) at the point v7, / of the dise, the
expression
z I a cos T7 cos acos 77 cos d7... (12);
V= 7r cos jji 28sin. Vnq, 7) l +a cos 0 tan (o)}d d..(2);
here the coordinates of the external point at which the potential is found are the
elliptic coordinates given by
z = r cos j0o, x = /r2 + a2 sin o0 eos o, y = /r + a2 sin Vo sin bo,
the coordinate 770 alone appearing explicitly in the expression. This formula agrees with
one obtained by Heine by a different and somewhat complicated procedure*.
* See his Kugelfunctionen, Vol. II. p. 132.