114 ELECTRICAL VIBEATIONS ON A THIN ANCHOR-RING [36!
and the value of the integral is accordingly
To this is to be added from (9)
a-o.2- I
4a
making altogether for the value of (8)
,,«>.
(12)
The second integral in (7) contributes only finite terms, but it is importan as determining the imaginary part of a and thus the rate of clissipatior We may write it
sin
{cos (2m + 2) -^ + cos (2m -2)^- 2
,... (13)
where x? = 4a2 a2 = 4m2 approximately.
Pocklington shows that the imaginary part of (13) can be expressed b means of Bessel's functions. We may take
2 r*77
- " d^r cos 2n^ eix sin * = J2n (x} 4- i Km (x\ ..........(14)*
whence I dty cos 2?ii^ .-=-^ {J"21l (x) + i Ifm (x)} dx......(15)
J o sin y* 2> J o
Accordingly, (13) may be replaced by
WI-TT fffi , y , /a.\_2J (afi + J dc^ + ifK -%K +K M C16") 4a JQ
JN OW J t72?n-|-2 2J gm -f- Jzm 2 == 4e/ 2,11;
r IK
SO unat I Ctfl/ (t/2?vi+2 ^"J2m ~T~ "sin 2} := ^^ am = ^^ami ""2»i+i......(1' )
. 0
The imaginary part of (13) is thus simply
.(18)
A corresponding theory for the K functions does not appear to have bee developed.
When m = 1, our equation becomes
fx-
_ | i incr =------ )./. lv\ .L (vM 4-
COS (x Sin -v|r) 1 ,
^~l log^ = -~{Ji(x)-J3(^} + --2
* Compare Theory of Sound, § 302.
......(19)
f Gray and Mathews, Bessel's Functions, p. 13.0) is