Malet — On the Negative Pedal of a Central Conic. 151 



XXIY. — On the Is'igative Pedax of a Centeal Cojtic. By John 



C, Malet, M. A. 



[Read, Februaiy 25, 1878.] 

 Abstkacx. 



Having in some preliminary investigations proved a certain property 

 of circular cubic curves which. I require for the direct object of my 

 Paper, I then investigate directly the principal properties of the first 

 negative pedal of a central conic from any point. Many of these pro- 

 perties I show are also true for a more general class of curves, viz. : 

 unicursal sextics -with six cusps : thus for any such curve the follow- 

 ing properties are true : — 



(1), The six cusps lie on a conic. 

 (2). The six cuspidal tangents touch a conic, 

 (3). The eight tangents at the four double points touch a conic. 

 (4). The six points of contact of the three double tangents lie on 

 a conic. 



I prove, however, many less general properties of the curve I 

 consider, which I believe are worth noticing — for example : — 



" If we take the fii'st negative pedal of a central conic from any 

 point on either axis, then the six tangents to the curve from the cusps, 

 but distinct from the cuspidal tangents, all touch the same conic." 



Again : — 



"The sixteen tangents at the eight double points of the nega- 

 tive pedals, with respect to the origin of the conies 



ac(? + hf + 2gx + 2fy + c = 0, 

 and 



5."/r + ((y"- + 2gx + 2fy - c = 0, 



all touch the same conic." 



The last part of my Paper is occupied with the consideration of a 

 curve which is the locus of the centre of a variable ch'cle, which cuts 

 orthogonally a given circle and touches a given curve. Prom the equa- 

 tion of this locus I prove that we may at once deduce the equations of 

 the following curves : — 



(1). The negative pedal of the given curve. 

 (2). The parallel of the given curve. 

 (3). The negative pedal of the parallel of the given curve. 

 (4). The locus of the centre of a variable cii'cle which touches the 

 curve and a fixed circle. 



I conclude by showing that wc can form the equations of the 

 parallel et cetera of a surface in an analogous manner. 



K. 1. A. I'llOC, SEK. II., VOL. Ill, SCIEXCEi M 



