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.

286 DR HOBSON, ON GREEN'S FUNCTION FOR A CIRCULAR DISC,
When the point P is on the axis of the dise, the induced charge is -q. -~, where 0o
is the angle subtended at P by a diameter of the disc.
When P is in the plane of the dise, the angle NPM becomes the angle between the
tangents frorn P to the circular boundary of the disc.
8. The surface density at any point of the disc is given by the formula
47r av
when dv is an element of normal and is given by
+ adO
dv = -
cosh p - cos '
We thus find for the density p, at the point (p, 7r,,) on the positive side of the dise,
=po =-4 * PQ3 {1 + -sin- (sin 2~ sech 2
cos O0
q 1 cosh p + 1 o
47r2 PQ a o1
4rcosh2 -Sa - sin2
this expression can be put into a more geometrical form by introducing the auxiliary
point L (po O-rr,,o) of Art. 6. The point L is now in the plane of the disc, and external
to the disc; denoting this position of L by Lo, its coordinates are p, O, b0,. We have
sin- (sin o 0 sech a) = tan-' (p~ A Q)
w, -, T(P. CL L-a2\
which is equal to -tan-' ( );
2- tn-~ P-Lo a2_
on reducing the second term in the expression for p, remembering that
a sin 0o
cosh po - cos o'
we find that it becomes
qz i /CL 2_-a2
-27r92 PQ2.PZo V 2 _ CQ2 )
2wr'PQ'.PLo a -CQ'
and thus the expression for the density at any point Q on the positive side of the
disc is given by
q PN q PNT PQ L- a' -a 2 (PQ CL'-' -a.,
po 27r PQ3 2 r2 PQ3 {JPLo a- CQ2 tPLV a- G } (7)
where PN is the perpendicular from P to the plane of the dise, and L, is a point on AB
produced, such that ALo: BLo= AP: BP.