1881.] of Cartesians and other Quartics. 83 



6. For x + iy = dn (£ + in), 



the foci are given by 



z=l, k', -k', -1; 

 and 



r 2 = dn (f + tiy) dn ((■ - irj), 



r l r i = k 2 sn (£ + ^) sn (f — t??), 



r 2 r s = k 2 en (£ + 177) en (f — 177) ; 



and therefore the quartic curves are given by 



r 2 r 3 — r l r i dn 2f = A; 2 en 2£ ) 

 r 2 r 3 + rj\ dn 2t?? = 7c 2 en 2in) ' 



with similar expressions connecting 



r 2 and r,^, r 2 and r 2 r 8 ; 



but these curves derived from 



x + iy = dn (f + iij) 



are similar to those derived from 



x+ iy = sn (f 4- in). 



These quartic curves obtained from 



x + iy = sn (£ + zt?) or en (£ -f i??) or dn (£ + it?) 



have been considered by Siebeck in Crelle, 57, and figures 

 given. 



The physical problems in connexion with the surfaces generated 

 by the revolution of these curves have been considered by Dr 

 Albert Wangerin in the Transactions of the Academy of Sciences 

 of Gottingen, 1875 ; and he introduces certain functions, analogous 

 to the toroidal functions employed by Mr W. M. Hicks for the 

 solution of corresponding problems for the anchor ring surface. 



Siebeck uses as vectorial co-ordinates the sum and difference 

 of the focal distances r 2 and r 3 ; and denoting them by S and D, 

 then 



r * = i(S 2 + D 2 )-l, 



rj' 3 = i(8*-D 2 ), 



and by substitution in the preceding equations his results may be 

 obtained. 



6—2 



