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. 281
coordinates po, 0+ 27r-, /b of the point P' in the second space, which corresponds to P.
If Q moves up to and ultimately coincides with P, we have cosh a=l; it will then
be seen that the first function becomes infinite at the lower limit, but that the second
one remains finite at that limit.
Consider then the function W(po, 8, b0o) given by
W(po 0o, 0o,) = 2 - (cosh p - cos 0)Y (cosh po - cos 0,)
sinh 2
f Vcosh u - cosh; 1
JaVcosh - cosha cosh U - cosh os - 0o)
the above equation may be written
- = (p, o0, o0) + W (po, 0 + 27r, 0bo).
PQ
It is clear that the function W is uniform in our double space as it is unaltered
by increasing 0 by 4rr; it will now be shewn that it is a potential function. We
nay express W in the form
W = (cosh p - cos 0)B (cosh p, - cos 0o) f 1 j + 2/e-4mu cos (0 - _Oo) du,
2 v27ra \a /cosh - cosh a 2
which may be written in the form
W= (cosh p - cos 0) (cosh p, - cos 0o,) Q_- (cosh a) + 2 Z Q (co ) cos ( - 0),
W7 = 'r5 P - - '
since the formula
- e-(n+~) u
V\-2 Qn ('cosh cc) e.-, ----, du,
2 Q (cosh a) cosh osh- cosh a)
holds for all values of n such that the real part of n + ~ is positive (loc. cit. p. 519).
Now (cosh p - cos 0)$ (cosh p, - cos 0,) cos s (0 - 0o) Qs (cosh a) is a potential form whatever
s may be, and thus W is a potential function, and is expressible in the form
W = - (cosha)2(cosh pa) cos (cosh p cos (cosh a) + 2Qo ( 0 - 0o)
+ 2Q~ (cosh a) cos (O - o) +..,
the value of W (po, o + 27r, o0) being
2vra '{
2- (cosh p-cos ) (cosh po- cos 0o)\ Q_ (cosh a) - 2Qo (cosh a) cos (0- 0)
+ 2Q_ (cosh a) cos (- 0) -...};
the two expressions added together give the expansion of l/D obtained in Art. 2.
VOL. XVIII. 36