Prof. Sylvester's Astronomical Prolusions. 63 



combining which with any of the four couples of linear equations, 

 pA± \/tyqB + qC + l = 0, p'A± \/V<?B + 2'C + l=0, 



obtained by substituting for x, y the coordinates of the two given 

 points, we obtain six sets of quadruple solutions, making twenty- 

 four finite solutions in all. This result is in perfect accordance 

 with that which applies to the case of the tangents meeting at the 



focus; for when - is the square of the principal semiaxis in 



which the focus lies, we have already found eight solutions ; and 



when - refers to the other semiaxis, we have 



q -* a ^ e >- *J + <*± \/(s*-c*)(s f *-c*Y 

 which, considering c, s, <r given, leads to a biquadratic in s* which 

 serves to fix the curve ; and as there are four systems of values of 

 s, a arising from the permutations of the signs of p, p f , we thus 

 have four times four, or sixteen solutions over and above the pre- 

 vious eight, making twenty-four in all, as in the general case. 



We might generalize the problem otherwise by supposing 

 given, not the magnitude of a principal axis, but that of a 

 diameter through the intersection of the two given tangents ; 

 or, again, in quite a different direction by supposing three points 

 P, Q, R to be given in a Cartesian oval defined by the equation 

 kp — p'=z2a, p referring to a given focus P, and p' to a second 

 focus G to be determined, a also being given, but k being to be 

 determined. It is easy to see that in this case also the position 

 of G may be obtained by the intersections of circles ; for the ratios 

 PG : QG : RG will be known ; there will thus be eight pairs of 

 solutions arising from the permutations of the signs of p lt p if p 3 



PG 



which measure PP, PR, PR ; and calling — e, it would be an 



interesting analytical question to express the eight systems of k and 

 e in terms of p v p 2 , p 3 , and c v c 2 , c s the three chords joining 

 P, Q, R, — these six quantities, of course, being not independent 

 but connected by the well-known equation between the mutual 

 squared distances of any four points from one another on a plane. 

 Touching the Cartesian ovals, Mr. Crofton has well remarked 

 that a circle may be regarded as one of a very peculiar kind. 

 Por if we take any two points electrical images of one another, 

 inverses, in Dr. Hirst's language, or, as I prefer to call them, 

 reciprocals or harmonics in respect to a given circle, the dis- 

 tances p, p 1 of any point in the circle from them will be con- 

 nected by the equation •— kp + p' = 0; so that any pair of 

 harmonics whatever of a circle may be regarded as foci of such 



