64 Prof. Sylvester's Astronomical Prolusions, 



curves. The third focus correlated to each pair will evidently 

 be the centre ; for, calling its distance from any point in the circle 

 p", we have ,p + p" = c; in the first equation the modulus k is 

 finite and the constant zero ; in the second the modulus is zero and 

 the constant finite. Consequently a circle is a Cartesian oval, 

 not only as a particular case of a conic, but proprio Marte 

 and porismatically in quite another sort of way *. Now it is 

 well known that a conic may be described by two forces (varying 

 as the inverse square of the distance, and tending to its two foci) . 

 This led me to inquire whether some analogous theorem did not 

 hold of a circle in respect to any of its pairs of foci, i. e. of har- 

 monics ; and I find such is the case, as the annexed simple inves- 

 tigation will make manifest. 



Call the radius of the circle unity ; c 3 - the distances of two 



harmonics from its centre ; ^» ^t- two forces tending to these 



p p n 



points respectively ; then by duly assigning the initial velocity, 



we are at liberty to suppose the constant zero in the equation 



for vis viva, so as to be able to write 



we have also 



(n— l)^- 1 (»—l)p / »-i' 



1 



-1 , 

 P _ c _ *■ 



p 1— c "~ c 



In order that the circle may be described under the circum- 

 stances above supposed, it is necessary and sufficient that 



v*=^cosi+£- n cosi', 



* Thus it will be seen that, besides its derivation through the ellipse, 

 the circle descends by a short cut immediately from the Cartesian oval ; 

 recalling to mind the singular condition of consanguinity of the ill-fated 

 descendants of Laius, at once children and grand-children to their mother, 

 sons and brothers to their father. Viewed as sprung from the ellipse, there 

 should be but two coincident Cartesian foci to the circle ; it is the fraternal 

 aspect of the relationship which brings into view the existence of an infinite 

 number of such foci in the circle ; every point in fact being a focus. This 

 is explained by considering the circle so descended, not (like a conic) as a 

 Cartesian oval with a branch at an infinite distance, but without such 

 branch, and as doubled upon itself; thus the circular points at infinity 

 become each double points, and, as well remarked by Mr. Cayley, every 

 line through either such double point is analytically a tangent to the curve, 

 and thus every point in the plane of the circle, being the intersection of 

 two such tangents, ought to be, as it is, & focus. 



