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.

WITH APPLICATIONS TO ELECTROSTATIC PROBLEMS. 287
The value p, of the density at the point (p, - r, b) on the negative side of the dise is
found in a similar manner to be
q PN PQ CLo/ -a' 1PQ /CL2 -a2- )
2P 2 P=Q 2 jPLO a' - CQ' ta PL 2 c...... ( 8)
Thus the densities at corresponding points on opposite faces of the dise satisfy the relation
q PN
Po- Pl= - PQ3
When P is on the axis of the dise, L, is at infinity, and the formulae (7), (8) become
q PN q PN x PQ tnQ
po-2r ' PQ3 2w7.2 PQ3 IAQ BQ tan A.BQ
q PN PQ tan- PQ \
P-= 27r {Q /nAQ.BQ tan- (AQ BQ) B(9)
The expressions (7), (8), (9) agree with those obtained by another method by Lord
Kelvin *.
When P is in the plane of the disc it coincides with L,; in this case we find
that the density on either side of the disc is given by
1 /CP2 a2
P= 27r2Q2 a2-CQ2............................ (10).
9. If the influencing point P is on the axis at 0o, we find from (5) the following
expressions for the potential at points on the axis:-On the positive side of the disc
q q (0-0,)- q2 ( +,) when 0>00,
PQ 2wr. PQ 2,. P'Q
q + 2 Q (0-0)- pQ (O + 0o), when 0 < 0.
PQ 2r.PQ > 2r. P'Q
On the negative side of the axis
-Q+ PQ q( 0) + (2 + 0,), xvhen 0 + 0o is positive,
PQ 2w. PQ (-0) 2r. P'Q
PQ+- PQ (0 - 0)- q (0+ 0,), when 0+0, is negative.
PQ 2w. Q( 2Pr.p Q
If we denote by z, the distance of P from the disc, and 'by z the absolute value of
the distance from the disc of a point Q on the negative side of the disc, the potential
at Q is given by the expression
q q q z _ q t o
-t-X + Z ct-l (z +- C ot-l - o t-l ~;
z+~z See his a)er on "z M-zl Ma" a
* See his papers on "Electrostatics ancd Magnetism," p. 190.