SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 
The ring P.F. of a circular element is 
(1) rfP= PQ'" = A»+2A// cos « + (/“ + ?/, 
and the S.F. is 
(2) dP = ^irtT dij [?/Q (MQ') —P7>]. 
Changing’ from Q (MQ') to the form 
o/r)' 7 \ f Aa cos 0 +A^ + /P hdO 
(3) Q(PZ)-J 
PZ^ = A^siiP(I + //, 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 dB 
(‘A.M.S.,’ p. 506, § 48), 
(4) 
P = 'liro-ydy 
r Ay cos 0 + A“ + h' h dB 
J A^sin^O-f-6^ PQ 
A^ cos^ B hdB 
A^'siir0 + 6-’ ^ 
A“ cos^ B + Ay cos B 
A'sin^0 + /P 
Ay cos B h dB 
A^ sin^hT^^' PQ' 
or with (T = hif. 
(5) 
P = 
r SttA^ cos^ 6 7 7^ ^hy'^^^dy 
'PQ' JA^sin^0 + // J PQ' 
27rA cos B 
A^ siiF B + b- 
h dB. 
Here the y integrations are effected by tlie formula of reduction oljtained from the 
integration of 
(6) ^(y’‘+TQ') = [(n + 2)y”+' + (2H + 3)Ay”+ico80 + (/i+l)(A^+6QyQ^, 
and so Pi, can be obtained in finite terms. 
But if we attempt the determination of the ■ P.F. the intractable 1 puts in an 
appearance when n is odd. 
Consider, for example, the flat lens of § 16, ‘ A.J.M.,’ 1919, where a- = k[l — 
\ Ct> ! 
( 2'' — o 
or for a- = ki I — , as in the distribution of electricity in the circular disc. 
\ a-/ 
21. Taking the form in (3), § 20, it cam be resolved into 
(1) ii(PZ) = -iA(PZ)+iA(PZ), 
/o') o —if a — ^/{A- + b'-) • bdB < , a+ \/{A^ + b-) b dB 
- 2 J + pQ> J y(A^ + hQ-AcosO' PQ ' 
two III. E.I.’s in the form of B in (lO), (ll), § 4. 
K 2 
