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VII. — On Bicikctjlar Quartics. By John Casey, A. B. [Abstract.] 



[Read February 10, 1867.] 



If we take the most general equation of the second degree in a, (3, 7, 

 where these variables denote circles in place of lines 



0, h c,f, g, h,) {c, /3, 7 ,) 2 = 0, 



we get the most general form in which the equation of a bicircular 

 quartic can be written. 



Setting out with this equation, I have proved that a bicircular 

 quartic is the envelope of a variable circle which cuts the Jacobian (*/") 

 of a, /3, 7, orthogonally, and whose centre moves on a given conic F; 

 the equation of the conic F in three point co-ordinates being exactly 

 the same in form as the equation of the quartic, the a, /3, 7 of the 

 quartic being replaced by X, /u, v of the conic, where X, fi, v are the 

 perpendiculars from given points on any variable tangent to the conic. 



I have further proved that the same quartic may be described in 

 more ways than one, in this manner, according to its class. Thus, if 

 the quartic be of the eighth class, there are four conies, F, F', F", F"\ 

 and corresponding to them four circles, J, J 7 , J' f , J'" ; and the same 

 quartic may be described indifferently as the envelope of a variable 

 circle whose centre moves along any of these conies, which cuts the 

 corresponding circle orthogonally. 



I have proved that each of the four circles, J, J', J", J" f } inverts the 

 quartic into itself. 



If the quartic be of the sixth class, there are but three director 

 conies, F, F r , F"\ and three circles of inversion, J, J', J". In this 

 case I have proved that the quartic must be the inverse of an ellipse 

 or hyperbola, being the one or the other according as the double point 

 it must have in addition to the circular points at infinity is a conjugate 

 point, or a real double point. 



If the quartic be of the fifth class, I have proved that it must be the 

 inverse of a parabola ; that it has but two director conies, F, F' } and 

 two circles of inversion. 



For the quartics of each class, I have proved that the conies, F, F', 

 &c, are confocal, their common foci being the double foci of the quartic ; 

 and that their points of intersection with their respective correspond- 

 ing circles, J, J f , &c, are the single foci of the quartic ; so that the 

 sixteen single foci of a bicircular quartic of the eighth class lie in fours 

 on four confocal conies, whose common foci are the double foci of the 

 quartic. 



The conies, F, F', F", F fr ', which, on account of the property just 

 stated, I have called the focal conies of the quartic, are intimately con- 

 nected with the whole theory. Thus, if F, F', &c., become circles, the 

 quartics become Cartesian ovals ; and if parabolas, the quartics reduce 

 to circular cubics. 



I have discussed Cartesian ovals from a new point of view, and have 

 entered rather fully into their properties. Thus, being given two circles, 



