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