1915] THE THEOBY OF THE HELMHOLTZ EESONATOR
and accordingly
371
For the values of ^ in (15) we now have with restoration of Jc
X (outside) = —-
log sin ^6 + log (1 + sin
W.......(36)
(inside)
- log (•£ sin 0) + log tan | (TT — 0)
#ca 5 ^ 175 i 1
These equations give the value of ty at any point of the sphere, either inside or outside, due to a normal velocity at a single point, so far as /c2c2 inclusive. The inside value is dominated by the term. — 3/&'-c2, except when d is small. As to the sums in 7c2c2 not evaluated, we may remark that they cannot exceed the values assumed when 0 = 0 and Pn (//,) = 1. Approximate calculation of the limiting values is easy. Thus
= - 0-79040 + 1-64493 - 1-20206 + 1-62348 = 1'2759 *.
In like manner 00 2« '-4- 1
* ChrystaPs Algebra, Part n. p. 348.
t 1917. Mr White has shown that the accurate value of the first sum is
and that of the second sum
7T2
8 8
-+-
il!-
3 6 '
so that for the two taken together as in (38), we have
A + 1^2 = 2.39292.
The coefficient of W jn (39) is then
1
Further in this equation
175
-1 + 2-39292 = 1-39863.
2 4--22...................^