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. 279
cosh p + cos 0
whence we find CP'2 a2 p + cos O
cosh p - cos O'
~~~~hen~ce ^2 = 2ya sinha p
~henc~~e ~~ = (cosh p - cos 0)2
thus f, z are expressed in terms of p, 0 by means of the formulae
a sinh p a sin 0
cosh p - cos 0' cosh p - cos O'
2. To express the reciprocal of the distance D between two points (p, 0, a) and
(po, 00, bo), we substitute for e, z and io, z, in the expression
1
- = {(z - zo)2 + 2 + o2 - 2o cos (s - o)}-f,
their values in terms of p, 0 and po, 0o; we then find
i _ 1 (cosh p - cos 0)3 (cosh po - cos 0o)2
D a V2 {cosh a - cos (O - o0)}
where cosh a denotes the expression cosh p cosh po - sinh p sinh po cos (b - bo). If we
suppose the expression {cosh a - cos ( - o)}- is expanded in cosines of multiples of
2 cosr os r7-s
0-0o., the coefficient of cos, ( - 0o) is - -h djvr which is equal* to
vr o (cosh a - cos r)2
7/'
2 -- Q_ (cosh a) when Qm-o denotes the zonal harmonic of the second kind, of degree
i 1 i
m -; thus = (os p - os 0) (coshpo- cos 80) E 2Qm, (cosh a) cos m (0 - 0o), where
the factor 2 is omitted in the first term, for which vm=0. The series in this expression for 1/D may be summed, by substituting for Qm_ (cosh a) the expression
i rf e-mu
/12 J (osh - -cosh d' (loc. cit. p. 519);
2a (cosh u - cosh a)'2
we find
1 = 1 _ (cosh p - cos 09) (cosh po - cos 02fc - 1+ 2Ze-nb cos m (0 - 09o)} d,
D 'ra 2J (cosh u - cosh a)'
and thus we have the formula
1 1 "'O.i/x i\ 0 1 sinh zu
1- - -a (cosh p - cos 0)' (cosh po - cos 0) - o h sinh du,
D 7ra v\2 a /cosh u- cosh a cosh u- cos (0 - 0o)
where a is given by
cosh a = cosh p cosh po - sinh p sinh po cos (b - bo).
* See page 521 of my memoir "On a type of spherical harmonics of unrestricted degree, order, and argument,"
Phil. Trans. Vol. CLXXXVII. (1896) A.