1881.] of Cartesians and other Quartics. 81 



and denoting by i\, r 2 , r s , r 4 the distances of a point from the four 

 foci A, B, C, D, given by z = a, b, c, d; then 



r, 6— d ,. 



r aZTd = sn (£ + lr >) sn (£ - lr >)> 



r a-b f 



r a -d = Cn ^ + ^ Cn ^ " *^' 



- 9 ? — " = dn (f + **) dn (£- ^) ; 



/ 2 U — L 



f n (i v a J) 



and consequently, writing in equations (A), — -7 for r, — - — -7 



V „ G 0/ V- Oj — Qj 



for r , and 



1 r. a — b r. a — b b — d „ ,, 



72 or -= ~ 7 , for r , 



k r 2 a — c r. 2 a — db — g 



the vectorial equations of a system of quartics with four collinear 

 foci may be written 



r 4 (a-b)- r, (b - d) dn 2£ = r 2 (a - d) cn 2£ 

 r 4 (a — b) + r, (6— ^) dn 2 177 = r 2 (a — d) cn 2?7i 



with similar expressions connecting 



If a = b and c = d, then 



^ — |(a + c) = i (a — c) coth \ u, 



giving the dipolar system of circles. 



4. The integral in its canonical form 



dz 



I 



V(l-sM-fcV) ' 

 gives z = sn v, 



or x + iy = sn (£ + 177) ; 



and the four foci taken in order are given by 

 -- 1 1 -1 - 1 



VOL. IV. PT. II. 



