SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 
57 
16. The S.F. of the P.F. W is L, so that the S.F. of P is 
da ’ 
and in (2), § 2, with 
( 1 ) 
( 2 ) 
(3) 
rlL 
da 
d h _ Acos6 + a h 
da MQ^ ‘PQ’ 
fo A fo A „AcOS0 + a 
j2xAcoseth ‘ PQ ^0- J cos 9 
_ T f o A^a^ cos^ 6 + Aa^ cos d bdd 
= -27r 
I AW + Aa^ cos Bh do 
= — 'iivcd' 
MQ^ PQ 
MQ^ —Act cos B — od' h dB 
MQ^ 
= — 27rPCT6 + 27rCT^Q (MQ), 
PQ 
clL 
da 
— 27rCT (2’7r — Q)—27rP6 = 27rCTt} (1 —jf ), 
is the S.F. of P, the rim potential of the circle AB, and L in (12), § 8, is the S.F. of 
the circular disc on AB. But then, from (8), § 13, 12 (l—/) f^e solid angle of the 
circle on the radius NP seen from Q on the circle on AB. 
The S.F. of I, P.F. of the cylindrical skin, is then given by 
(5) 
J = 
^ ^ dh = 27rCT 
da 
— MQ^ + Act cos B + 
CT 
MQ^ 
PQ dB 
— ttct I ( —MQ^+ct^—A^) (14- 
dB 
MQV PQ 
-7rCTj( MQWct^ W ?>^)^^+7rCT6j • PQ 
= ttct |( —2 A^—6^—2 Act cos B) +47rCT6B 
= -7rP(2AW6 ^)-MCT + 27rCT612(MQ)-7rP6^ 
= -27rP(AW6^)+27rQACT + 27rCT6(27r-12), 
( 6 ) 
aL_Wct= -P(CTWAW6') + 2QACT + 27rCT6, 
2'7r 
so that J is given in finite terms, while I is intractable, and requires to be given in a 
series. And generally in these investigations we find the S.F. has the superiority 
over the P.F. in simplicity of analytical structure. Thus the S.F. at P of the rod AB 
is PA —PB, and of the electrified disc AB is ^/[AB^ —(PA —PB)^]. 
