VOL. XXIV.] PHILOSOPHICAL TRANSACTIONS. 153 



this curve, with some of their properties, not before sufficiently examined, 

 have occurred in my inquiries, as follows. 



It is well known that the nature of the curve is this: If from two given 

 points, p and g, fig. 1, pi. 6, to any point in the curve h, the two right lines 

 PH, GH be drawn; the rectangle under ph and gh is equal to a given space. 

 The right line fg produced both ways till it meet the curve, shows the two 

 vertices a and b ; and ab is the principal axis; also the middle point c between 

 the vertices is the centre of the figure; and de drawn through c perpendicular 

 to AB, is the less axis; and the points p and g are the two foci. 



In this figure, if the less axis be greater than the distance between the foci, 

 the curve terminating the figure is every where concave towards the centre, 

 such as the figure is commonly described. But if the distance of the foci be 

 lessened, while the principal axis continues the same, the less axis will be in- 

 creased, which yet remains less than the axis of an ellipsis, described with the 

 same principal axis and the same foci ; till at last, when the foci unite, it be- 

 comes equal to the greater axis, and the figure changes into a circle. But, on 

 the contrary, if the distance of the foci increase, the less axis will be di- 

 minished, and will become equal to the said distance, when this is to the 

 principal axis, as unity is to a mean proportional between 1 and 3. 



If the distance of the foci be further increased, the less axis will be still 

 diminished, and the curve at its extremities will no longer be concave towards 

 the centre, but convex, as in fig. 2, till the distance of the foci be so far 

 increased, as to be to the greater axis, as the side of a square is to its dia- 

 gonal ; then the less axis will become nothing, and the curve touch the centre 

 on each side. 



If the distance of the foci be greater than in the aforesaid ratio, the 'ess 

 axis becomes impossible, and the figure changes into two conjugate figures, as 

 in fig. 3, which will be diminished as the distance of the foci increases, till at 

 last the figures vanish in two conjugate points only. 



The distance of the foci still increasing, the two conjugate figures emerge 

 again, which increase in the same manner as they before decreased, being dif- 

 ferent from the former in the order of the foci and vertices, and so proceed 

 increasing to infinity. And afterwards this system will again approach to the 

 circle gradually as before it receded from it. 



Hence it appears at first view, that this figure cannot at all be proper to 

 constitute the orbit of a planet. For, not to mention the case in which it 

 becomes two figures, and forsakes the nature of an orbit, viz. whenever its 

 eccentricity is so great as the comets require (if they move round the sun like 



VOL. V. X 



