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.

284 DR HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC,
hence
sin- {cos (0- 0,) sech 1 = + tan-1 (PL V/az- 1 );
p
p/
FIG. 2.
in order to determine the sign on the right-hand side, we observe that the inverse
sine is positive unless 0 lies between - (r - 00) and - r, that is unless Q lies within the
sphere passing through P and the rim of the dise, and is on the negative side of the
disc; thus the sign on the right-hand side is to be taken positive unless Q lies within
this spherical segment.
Similarly we find
sin-l -- cos 2 ( + 0,) sech a = t tan-l (Q / 2 - )
where the negative sign is to be taken unless Q is on the positive side of the dise and
within the sphere which contains the rim and the point P'. We have thus as the
expression for the potential of the system at any point Q (p, 0, b)
V= Q L1 + -tan-(' v F tan-2 C.) (4)
P _PQ - CL 2-a/2 2P 1 ~Q.2 'tan-PQ /CL-.(2\a
when the ambiguous signs are assigned in accordance with the above rules.
The auxiliary point L may be found from the following construction:
Draw a spherical bowl through the rim of the dise on the opposite side to that on which
Q lies, and equal to a sirnilar bowl which passes through Q; draw a plane PA'B' through
P and the axis, cutting the rim in A', B'; this plane intersects the bowl in a circle; on
this circle L lies, and is found by taking it so as to satisfy the relation
LA': LB' = PA': PB'.