SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 61 



The ring P.F. of a circular element is 



and the S.F. is 



* 



(2) dR = 2 dy [yQ (MQ')-Ffc]. 



Changing from Q (MQ') to the form 



/, /~>/T>r7\ f A cos + A 2 + & 2 bdO 



pz , ""PQ ' 



PZ 2 = A 2 sin 2 $ + & 2 , as not involving a or y, PZ the perpendicular on OQ, this form 

 of Q (PZ) is obtained by the dissection of the circle into the sector elements Ja 2 d0 

 ('A.M.S.,'p. 506, 48), 



t ,\ Co j f A/ycos fl + A 2 ^-^ 2 bd6 f, 7 7 [ydQ 



(4) R = | a^cfr J.-t__ . - j 2ft4f J t_ 



ff , , A 2 cos 2 fl />dfl |T i Ay cos />d0 

 = J J 3TOy ^ A' sin' + 6* ' PQ 7 + JJ '^ dy A-8infl + 6- ' PQ 7 



ff , 7 A a cos- 6 + Aw cos bdO 



= ZTTO-W ttW r - -FT7T/ ' 



JJ A 2 sm 2 e + // PQ' 



or with <7 = ky", 



tky*+ l di/' 2xA 2 cos-0 ,,. , f ky n+2 dy f 2-TrA cos , , fl 

 R = J PC . A 2 sin 2 e + ft 6 rf + J PQ^" J A 2 sin 2 6 + V ' '' **' 



Here the y integrations are effected by the formula of reduction obtained from the 

 integration of 



(6) -j- (z/" +1 PQ') = [(n + 2)?/" +2 + (2n + 3) Az/" +1 cos 6 + (n+ l) (A 2 + 6 2 )y"] -= ^ , 

 dy l Jr Q 



and so R can be obtained in finite terms. 



But if we attempt the determination of the P.F. the intractable I puts in an 



appearance when n is odd. 



/ ? ,- j \ 



Consider, for example, the flat lens of 16, ' A.J.M.,' 1919, where or = kl 1- ;); 

 or for or = k(l -^rj , as in the distribution of electricity in the circular disc. 

 21. Taking the form in (3), 20, it can be resolved into 



(1) ! 



i ( -V(A 2 +6 2 ) bde , f a+y(A 2 +6 2 ) bde 



2 '' 22 6 2 )-Acos0 PQ ' 



two III. E.I.'fl in the form of B in ( LO), (l 1 ), 4. 



K 2 



