52 
SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 
And similarly 
(13) 
(1-,)K' = 2GV, = 
as in the quadric transformation. Thus 
8ttK 4aK/c^ 
(14) 
P = 4Kdn/K' - 
or 
4GtG y 
r-i v/(^srs) ' 
On fig. 1, 0 = 0 + 0', 
(I5) cos d = cos 0 cos 0 '— sin 0 sin 0' 
(16) 
cn 2eK dn 2eK —/c sid 2eK, 
dQ = = dn fK' {k sid 2eK-cn 2eK dn 2eK) d2eK, 
PQ 
and integrating round AB from 0 < e < 2, the second term in dQ, vanishes by 
inspection, and 
(17) 
Q = 2+LA ('’(i_d„^2cK)rf2eK = * 
K Ji) 
^^^4(K-E) = (K-E)^, 
2 A 
\/{rd'-s) 
K 
K 
(18) 
M = -2xQA = -4x(r,+n)(K-E) = -2,rv/(»v,,) 
Thus in the construction of the curves of constant M on the Weir chart, a table 
was first drawn up from Legendre of E and K for every degree of the modular 
angle 6, and then of K — E and 
sin 
latitude, such that ch >; = sec X, and 7*2 — c (ch >/± cos ^), 
(19) N = - ch , (K-E), cos X = 
E 
0- 
and with the hour angle a = — and X the 
( 20 ) 
sin 6 = 
T3-^'2 
COS £ 
ch >] 
sin a cos X, 
sin a 
sin 6 _ N 
cos X iv —E ’ 
sin 6 
whence X, a were calculated for given N, starting from X = 0, when N = K —E, a = 6 
Another method is given by Maxwell in ‘ E. and M.,’ § 702. 
