52 SIR G. GREENHILL ON ELECTROMAGNETIC INTEGRALS. 



And similarly 



(13) . (1-)K' = 2GV, 1^2^, 



as in the quadric transformation. Thus 



.vi fvi 



(14) P = 4Kdn/K'= --^^ or 



On fig. 1, = + 0', 



(15) cos = cos cos 0' sin <p sin 0' 



= en 2eK (In 2cK-- sn 2 2eK, 



(16) dQ = - ac = dn/K' (,-sn 2 2eK-cn 2eK dn 2eK) d2eK, 



^ 



and integrating round AB from < e < 2, the secolid term in cZQ vanishes by 

 inspection, and 



(18) M = -27TQA = _4T(r,+r,) (K-E) = - 



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 



TT _ p 



angle 0, and then of K E and : - ; and with the hour angle a = -kir P, and X the 



sin 



latitude, such that ch >j = sec X, and r 3) r 2 = c (ch ; cos ), 



N= - 



/on\ 7 's~ r 2 c s sin 6 N 



(20) sin 6 - 2 = = sin a cos X, sin a = - - = * = , 



r 3 + r 2 ch,; cosX K-E 



sin 9 



whence X, a were calculated for given N, starting from X = 0, when N = K E, 

 Another method is given by MAXWELL in ' E. and M.,' 702. 



