1881.] of Cartesians and other Quartics. 91 



having four foci A, B, G, D forming a kite-shaped figure; and 

 then 



z — a a 1 — en u 



where 



and 



Then 



and 



z — c /3 1+cnw' 



a = J{{m-ay + n 2 } = AB or AD, 

 /3 = V{(w - cf + n 2 } = BG or CD; 



l.-i/i ■ AB* + BC*-AC 2 \ 

 16 -*[ L+ 2AB.BG J 



= i(l+cos ABC) 

 = cos 2 1 ABC. 



r t _ a en 177 — en f 



r 3 /3 en it; -f en £ 



r 3 _ dn £ dn ^ + *&&' sn f sn it? 

 r 4 dn £ dn i?; — ikk' sn £ sn 177 ' 



and by the alternate elimination of £ and nj, we obtain the vec- 

 torial equations of the orthogonal quartic curves, having foci at 

 A, B, G, D. 



When ABCD is a rhombus, the system of curves is as in § 5, 

 given by 



x + ty = cn (£ + ty). 



Generally if we invert the system of curves of § 10 with 

 respect to any point whatever, we obtain the second class of 

 bicircular quartics, in which the four real foci do not lie on a circle, 

 but are so related that 



AB : BD :: AG : CD. 



12. In order to solve the hydro-dynamical problems of deter- 

 mining the current and velocity functions of liquid motion due to 

 the motion of cylinders whose cross-sections are curves £ = constant, 

 or 77 = constant; taking for example the confocal Cartesians denned 



x + iy = m 2 (i; + iy) ; 



