300 Prof. Sylvester on Periodical Changes of Orbit, fyc. 



his beautiful paper on the Cartesian ovals. I find that when 

 the focal cubic is defined by means of the circle # 2 + ?/ 2 — c 2 =0, 

 and of its intersection with the parabola A# 2 + 2ex -f 2fy -\-g = 0, its 

 equation becomes Ae#(# 2 + y 2 -t-c 2 ) + (A 2 — Ag)x*- —\ey — jfe) 2 = 0. 

 I have already implicitly alluded in a preceding footnote, but 

 think it well again to call express attention, to the remarkable pro- 

 perty of the new ovals, of giving circular perspective projections on 

 the same plane for three different positions of the eye, the lines 

 joining the eye with the centre of each projection being all thjree 

 parallel to one another and perpendicular to the plane of the pic- 

 ture. This fact involves the truth of the elegant and probably 

 well-known elementary geometrical proposition, that if the oppo- 

 site sides of a quadrilateral inscribed in a circle be produced, the 

 lines which bisect the acute angles thus formed will be perpendi- 

 cular to one another, and respectively parallel to the two bisectors 

 of the angles formed by the diagonals at their intersection. I must 

 now leave to professed geometers (among whose glorious ranks 

 I do not claim to be numbered) the further study of those won- 

 derful twin beings, twisted Cartesians as I have called them, but 

 which those who so think fit may of course designate more simply 

 as ovals with the name of their originator prefixed. By suppo- 

 sing the vertices of the three containing cones to be brought 

 indefinitely near to the plane of the picture, my ovals ought to 

 revert to the Cartesian form. 



Woolwich Common, 



March 26, 1866. 



Errata in No. 206. 



Page 60, + cos i = cos \i cos <p, for cos p lege cos X. 

 s>pi+p 2 lege s^Cp l -}-p 2 . 



— 67, for ae— r lege ae= ~. 



° 5 



— 70, footnote, for M. Serret (bis) lege M. Ossian Bonnet ; 



and for centre of force ending singly lege centre of 

 force acting singly. 



— 71,/°,I = >4 = £, 



dr] lege P=|J dr\ . 



and/or P= ■£ f %»+»*)*• , p^gf" 2(p*+r>)rdr 



4 Jo (P 2 -r 2 ) 3 9 4j (pt-rV 



— 74, first footnote, for r' 2 = GO . GF lege r' 2 - GO . GF' ; 

 and for image circle lege image-making circle. 



