88 Mr Greenhill, On Conjugate Functions [Mar. 21, 



Then we may put 



z—ab—d „ .. . . 

 z — b a — c 



and therefore 



- JJ) = Sli (f + «M Sn (? - M/r 



?■ AB ,. . . Itm . . 



rjj> = cn (£ + *?) cn (£ - M >)> 



r 5 ZO = dn ^ + ^ dn ^ ~ ^' 



Therefore, as before, 



r^AE- r t BD dn 2| = ?>4i> cn 2£ 



?- 4 ^5 + r^D dn 2^77 = r % AD cn 2i7? ; 



the vectorial equations of confocal orthogonal bicircular quartics, 

 each of the form 



Ip + mp + np rr = ; 

 with similar equations of the same curves, connecting r t , r 2 , r 3 and 



r 2' T 3> V i- 



Hence the theorems are proved (Salmon, Higher Plane Curves, 

 Chap, vi.) that ' given four coney die foci of a bicircular quartic, 

 two such quartics can be described through any point, and these cut 

 each other at right angles." 



Also, if any cylinder, whose cross section is a bicircular quartic, 

 be electrified, the electrification at any point will be inversely pro- 

 portional to the square root of the product of the distances from 

 the four real foci. 



All these theorems are obtained immediately by the inversion 

 of the system of confocal Cartesians of § 2, three foci of the inverse 

 system of curves, the inverse points of the foci of the Cartesians 

 lying on a circle passing through the centre of inversion, the 

 fourth focus being at the centre of inversion ; and then the vecto- 

 rial equations are obtained by elementary geometry. 



