40 
SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 
3. In the reduction of these integrals to a standard form, for the purpose of a 
numerical calculation by use of the tables of Legendre, it is convenient to put 
0 = 2co, ft) = ABQ, in fig. 1, and to introduce a new variable t, and constants t, to, 
in accordance with the notation of Weierstrass, such that, m denoting a homogeneity 
factor, 
(1) PQ2 = ^ MQ^ = PM^ - = rri^ {t.-r), 
PA^ = 7-3^ = (b-L)) = ’’2^ = (b-b). 
MxA^ = = (rt + A)^ = nf{T — t^), = r.^—h^ — {a —AY = m?{T—tY), 
PA^—PQ^ = = 2Aa (l— cos 6) = 4Aa siid w = w? {t — tY), 
PQ^—PB^ = = 2Aa (l + cos O) = 4Aa cos^ w = 
PA-^-PB^* = 7’32-7-/ = 4Ag = m^{t,-tY), 
(2) 00 > > T > b > ^ > b > °°- 
In the notation of Legendre 
(3) 
(4) 
PQ^ = = i'Y cos^ ft)+ 7 ’/ sin^ co = {w, y), y' = ^ , 
Vo 
do 2d. 
(JO 
P = 
ado 2a ^ d. 
1<j0 
4aG 
PQ 7*3 Aft) Jo PQ 7’3 Jo Aft) 
and P is the rim potential of the circle on AB, with 
dw -r. 4a 
Vo 
(5) G = 
ia\ n P — a cos d r/o • 2 
P = 
0 A (ft), y) ’ y/ [r^rY) 
—a Q.OB 0 do T/,...- 2 , \ 2a c?ft) 
3 Aft) 
[2(1-AM-/]A 
y 7 3 Jo Aft) 
Gi\/ y. 
= A- [(2-/)G-2H], H = E(y) = f A(«,, y)do,, 
y'G L 
(7) 
M = - 27 rQA = - 27 rr 3 [( 2 -y^)G- 2 H], 
and Q cos <p is the magnetic potential of the circle on AB, with uniform magnetisation 
parallel to AB. 
In the Third Elliptic Integral keep to the Weierstrassian form, with the variable t. 
( 8 ) 
77^ 
dt = 2Aa sin 0 dO = 77r^y/(b—^ ^ —b) 
do 
2 dt 
h do _ 2v/(b —t) dt 
PQ “ mo/T ’ PQ “ 7T 
(9) 
