1912]
ELECTRICAL VIBRATIONS ON A THIN ANCHOR-RING
119
To obtain the mean over angular space we have to multiply this by sin 6d6, and integrate from 0 to \ir. For this purpose we require
f~Vw2(msin0)sin0d0, .........................(44)
Jo
an integral which does not seem to have been evaluated.
By a known expansion* we have Jo (2m sin 6 sin £/3) = J02 (m sin 8) + 2JV (m sin 0) cos /3 + 2 J22 (m sin 6) cos 2/3
-f
whence
.'0
Jo (2m sin 0 sin |/3) sin 6d6
= I "W Jo2 (m sin 0) sin 0d0+2 cos /S | ""' J? (m sin 0) sin 0d0 +
.0 .'o
+ 2 cos n/3
.'o
Nowf for the integral on the left
f *"" . sin
J0 (2m sin 0 sin ^/3) sin QdQ Jo and thus
4*
sin 0)sin0<i0.....................................(45)
---
riT
Jn2 (m sin 0) sm 0cZ0 =
7rt ğ d6 cos nB
cos
sin (2m sin ^)
'2m 0
......(46)
as in (15). Thus the mean value of (43) is
ra /*2ğ&
d% {J"2m+2 C53) + ^2m-a (f33) 2 J2m, (a?)
as before.
~
7/6
) =?^J
l-i(2m)-
In order to express fully the mean value of P2 + Qz + R2 at distance r, we have to introduce additional factors from (29). If a = a1- iaz, ei*r _ g^f.,1- eĞ#} aELCj t^ggg factors may be taken to be a4a2e2lx'r/r2. The occurrence of the factor e2"2'1, where a2 is positive, has a strange appearance ; but, as Lamb has shown J, it is to be expected in such cases as the present, where the vibrations to be found at any time at a greater distance correspond to an earlier vibration at the nucleus.
* Gray and Mathews, p. 28.
t Enc. Brit. "Wave Theory of Light," Equation (43), 1888; Scientific Papers, Vol. in. p. 98.
t Proc. Math. Soc. 1900, Vol. xxxn. p. 208.l's functions