90 Mr Greenhill, On Conjugate Functions [Mar. 21, 



(6 denoting the modular angle \BAG), 



and therefore 



_ . a k — ik en u 

 = 2xe-* e — — — ; 

 1 + en u 



, 2 _ a 2 k'—ikcn(f; + in) k' + ik en ((• — itj) 

 2 1 + en (£ + it]) 1 + cu (£ — 177) 



2 /'dn f dn ir\ + ikk' sn £ sn irj 



en 277 + en £ 



_ dn £ dn mi + ikk' sn £ sn in 



or r„ = 2a — — — ■ s- 5 



en %7\ + en £ 



and similarly 



_ _ dn £ dn iv\ — ikk' sn £ sn it) 

 3 en ^77 + en ff 



r. = a 



Expressed in a real form 



1 — en £ en 97 

 1 + en f en 97 ' 



_ dn £ dn 77 — kk' sn £ sn 77 



2 1 + en f en 77 



_ dn £ dn 77 + kk' sn £ sn 77 

 r=2a — * = 2 ^; 



3 1 + en £ en 77 



ot — 7* 



or 1 = en £ en 77, 



a 4- 7-j 



r f^ = cn(K-^)cn(K'- v ); 



' 3 ■" ' 2 



the elliptic functions of 77 being now to the complementary modulus 

 k', and by the alternate elimination of £ and 77 we obtain the 

 vectorial equations of the orthogonal quartic curves, having foci at 



a, b, a 



11. Inverting the previous system of curves with respect to 

 any point on the line through A perpendicular to BG, we obtain 

 the system derived from the integral 



dz 



J \/{z — a.z — c.(z — m) + ?i 2 



