58 HIK G. GREENHILL ON ELECTROxMAGNETIC INTEGRALS. 
17. The P.F. of the solid cylinder is given by 
( 1 ) 
in which 
( 2 ) 
(3) 
(4) 
Vadh = 
V = j WcZ6, 
= { th-i ^ 
do dh 
PQ 
— Q A = J j Ka cos 6 
PQ 
do dh 
w 
do = a\, 
L 
27r’ 
■Qb dh = I (M^Q) — 27r] 6 dh 
Aa cos 0 + a^ 1/ dO dh 
MQ^ PQ 
I 
— irV, 
bringing in again the same intractable integral I. 
We obtain *V otherwise from the integration of 
( 5 ) 
dV 
da 
= I. 
As a verification we have to prove by difierentiation that 
( 6 ) 
1 dJ ^ c?l ^ p 
27rA cZA dh 
implied in the integration of (l), § 15, and 
(7) 
—:rr = — 'ZttA — —— = —27rPZ> + 27rCfc (271 Q) = 27rttt} (l— J'), 
dh dA da \ ^ / 
implied in (4), § 16, the expression of the rim S.F. of the circle AB. 
dL 
And for the P.F. W and its S.F. 
da 
( 8 ) 
(9) 
2xA ^ = -2 xQA Ah 
dA da dh 
-27rA^ = 27rl}A = 
dh 
da dA 
18. An integration of L in (6), § 2, with respect to h will give the S.F. of the solid 
cylinder 
( 1 ) 
N = 1 Ld6 = K f 
+ A^ — 2 Aa cos 0 — ^ 
J J 
L MQ" J 
PQ do, 
