SIR G. GEEENHILL ON ELECTROMAGNETIC INTEGRALS. 63 



We may drop G' without ambiguity, and then 



(17) en/ dn/, = 



on f dn / -- V / (^ 



CI1 /2 all ./3 - 



/ JP f \ _ a I , -an/ cn/ 2 dn/ 2 _ 6 

 (t/3 ^^ ^ 2 



_ n 



(20) sn (/. + /,) = = r ) sin BOP = g| sin BOP 



UrJ 



OBF = am (/,+/) G' = am 2/'G', suppose. 



When OP is produced to cut the circle on AB in R, and the circle on ED again in 

 P', PP' will touch the co-axial circle in B, ; and by the poristic property of these 

 circles with the elliptic function interpretation, 



(21) OBP t = arn/G', OBP a = am/ 3 G', 



if the tangent at the lowest point e, e' of the li circle, and of the other co-axial circle 

 touching PP", where EP" = EP, cuts the circle on ED in P, and P,. 

 22. Treating Q (PR) of (3), 9, in the same way 



(l) ' 



, r vV+z^-A bde i f yV+frp+A bde 



* } y( 2 + fe 2 ) + cos e PQ ' 4 ~ * J v/( 2 + ^ 2 ) -a cos PQ 



and a similar reduction will give 



(3) Q 3 = f ^=^. 



J T 8 t 



(4) Q 4 



(5) 00 > t, > T- A > t. 2 > t 



