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Full text of "Scientific Papers - Vi"

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  ""2i+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