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. 285
In the case in which the influencing point is on the axis of the disc, we have po=O,
hence a= p, and the auxiliary point L is on the axis of the disc at the point where this
axis is cut by the sphere through the rim and the point Q, on the opposite side of the
disc to Q; the formulae for the potential then become
V=Q + - sin-' cos (O - O) sech p }] - P + -sin- -cos (0 + 0) sech }
q 1ta- __ /a2 - CQ' qP'L a - CQ2
q L1 + an-lP C - + tan-
C2PQ - 7. P 017 - a)] 2P' Q 7 P (Q CL - a'] ( 5.;
the sign in the first bracket is positive unless Q lies in the segment ApB, and the sign in
the second bracket is negative unless Q lies in the segment Ap'B.
7. To find, in the general case, the induced charge on the disc, it is sufficient to
examine the limiting value of the potential at a point Q, as Q moves off to an infinite
distance from the disc in the direction of the axis. In the expression for - q. GQ given
by (3), let 0=0, p= 0, then a =p,, and PQ, P'Q become infinite in a ratio of equality;
the expression for the potential of the induced electrification on the disc bas therefore
the limiting value
- 2q I i
o Q s- c os -o 00 sech -2P,
therefore the whole charge on the disc is
- q. - cos- (cos2 0 sech o)
which is equivalent to
_ 2 t-,(>/a-CL )
- tan-'
~ q PL;'
when L is a point in the plane of the dise which lies on the bisector of the angle APB.
This expression may be interpreted thus:p
FIG. 3.
Let PL be the bisector of the angle APB, draw the chord NLM perpendicular to
AB; the total induced charge is
L NPM
-q......................................(6).
q'F