G4 
Sin G. GREENHILL OX ELECTROMAGNETIC INTEGRALS. 
( 6 ) 
(7) 
sn 
sn/Xr = + A ^ ^ 
-V Jid^ + lA+a" 
r. 
v/(fX + />^)+A’ 
+/>^) + + />“)+ft 
and so on, with = 2 /. Because 
= y' sn(l-/;)G', 
( 8 ) 
(9) 
( 10 ) 
( 11 ) 
/ r v;- \/(ft®+/X) —A 
cn/; dn./; 
2 Ah 
cn/l dnXl 
b 
2Kb 
[_s/{aKb"^) d Kf 
sii {fK,Q = - = sn 2/G', 
^ Q 
sii U\-f^) = 
A6 
A 
sin OBF 
’*2 v/(«■ + 
Produce NP parallel to AB to cut the circle on ED again in P 4 , then 
n9l A NP BP . 
^ ' v/(«'' + /d) NB BP 4 ’ 
because P, P 4 are inverse points in the circle, centre N, through B; so that 
(13) 
(/<-/,) = ll-sin BPP, = sin BP,P, 
(14) 
BP. 
BP 4 P = am ifi-fs) G', OBP 4 = am ( 2 -/ 4 +/,) G'. 
Thence a geometrical construction for am f^G' and am./ 4 G', similar to that above 
for/i and,/;. 
Tlie pole of the cliord B.B,' througli B will lie on the line through A perpendicular 
to AB, at A' suppose, and the tangent A'B' will be parallel to AB. 
A whole chapter might be written of elliptic function theory, showing in this 
manner the geometrical interpretation of the various formulas, especially of the 
quadric transformation, in relation to co-axial circles. 
23. Our chief object was to employ a straightforward integration of Maxwell’s 
result as a direct road to the analytical results required in ampere-balance current 
weighing. The check on the arithmetical calculations has been explained and carried 
out in the ’Transactions of the American Mathematical Society’ (‘A.M.S.’), 1907, 
§ 56, p. 516. 
Considering that the chief analytical and numerical difficulties in these operations 
arise in the TIT. E.I. expression of ii, and that tins is cancelled by making 
( 1 ) 
A = 
ft. 
y — 12 = X 
— 2 Gy^ = 71— 2 K ( 1 —/c). 
