1881.] 



of Cartesians and other Quartics. 



79 



r — cn (£ + 4*77) cn (£ — ^77) 



_ cn 2t'?7 dn 2£ + cn 2£ dn 2/77 

 dn 2x7] + dn 2£ 



_// 2 dn2^-dn2 ^ 



~ A; 2 cn 2irj dn 2£ - cn 2f dn 2^ ' 



1 cn 2z?7 dn 2£ + cn 2£ dn 2^ 



F 

 F 



cn 2i?7 + cn 2£ 

 cn 2i'r; — cn 2£ 



& a cn 2i?7 dn 2£ - cn 2£ dn lir] ' 



and from these conditions 



r — r dn 2£ = cn 2£ ) 

 r + r dn 2ir) = cn 2^ j 



r" — r cn 2£ = p dn 2£ 



r" + r cn 2177 = y- 2 dn 2t?7 

 A/' 



r" dn 2£ - r' cn 2£ = 



'2"1 



or 



r" dn 2ir] — r cn 2irj = -=-, 



&'V-r'dn2£ + #V"cn2£ =0 

 — k'~r — r dn 2^ + &V" cn 2irj = 



(A) 



the vectorial equations of one and the same system of confocal 

 Cartesians. 



Since 



dz 

 du 



= 2 sn u cn u dn u, 



therefore if J denote the Jacobian, 



d (x, y) 



J= 4 sn (£ + iv) cn (f + irj) dn (£ + irj) 

 sn (£ — 1*77) cn (£ — ^77) dn (f — ?'?7) 

 = 4<k i pp'p". 



