1881.] of Cartesians and other Quartics. 87 



12 12' ^<! "*^ 



tin u tin u = -\ 



r* (in — m'y + (n + n'f 



n r* 



r 

 or sn u sn it = 



_ i 



en u en w 



and therefore 



1 ~ kr 2 \/ n" 



, , , &V, In 

 an it an w = — - A / — ; 



?* 2 V w 



k'r 4 fj-, - r t dn 2f = fcr, en 2£ 



/" , • P 



&V 4 a / — +r, dn 2t?/ = ftr 2 en 2it; I 



with similar expressions connecting r„ r 2 , r 3 and r 2 , r 3 , r 4 , are the 

 vectorial equations of orthogonal quartic curves, with foci whose 

 co-ordinates are 



(m, + n) (m, ± n). 



9. Generally, if a, b, c, d be the vectors of the four foci 

 A, B, C, D, then in the integral 



r dz 



) si (z — a . z — b . z — c.z — d)' 



the modulus of the elliptic functions will be real, when 



b — c a—d 

 a— c b — d 



is real ; that is when the angle between the vectors AD and DB 

 is equal to the angle between the vectors A C and CB ; that is, 

 when A, B, C, D lie on a circle ; and then 



BG AD 

 k ~AGBD' 



the anharmonic ratio of the four points A, D, C, B with respect to 

 any other point on the circumscribing circle (Salmon, Conic Sec- 

 tions, Chap. ix). 



