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.

278 DR HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC,
P moves from an infinite distance along a line above the plane of the circle up to
any point inside the circle, and in its plane, 0 is positive and increases from 0 to 7r,
whereas as P moves from an infinite distance along a line below the plane of the
FIG. 1.
circle up to a point within the circumference, 6 is negative, and changes from O to
- 'r. The coordinate qb is restricted to have values between O and 27r, and the coordinate p may have any value from - o to +oo, which correspond to the points A, B
respectively. The system of orthogonal surfaces which correspond to these coordinates
consists of a system of spherical bowls with the fundamental circle as common rim, a
system of anchor-rings with the circle as limiting circle, and a system of planes through
the axis of the circle. If we denote by e the distance CN of P from the axis of
the circle, and by z the distance PN of P from the plane of the circle, the system
Icos 5, sin (, z will be a system of rectangular coordinates, which can of course be
expressed in terms of p, 0, b. Let the lengths PA, PB be denoted by r, r' respectively, then r/r' = log p; we have
2rr' cos 0 = r2 + r'2 - 4a2 = 2rr' cosh p - 4a2
2a2
hence rr = 2a
cosh p - cos O
Again, z. 2a = rr' sin 0,
a sin 0
hence z= -
cosh p - cos O'
also since r2 + r'2 = 2a2 + 2CP2,
we have CP2 = rr' cos 0 + a2,