374 PHILOSOPHICAL TRANSACTIONS. [aNNO I7I8. 



To find the Curve which a Descending Body describes in the shortest Time ; being 

 urged by a Centripetal Force tending to a Given Point, and increasing or de- 

 creasing according to any Power of the Distance from the Centre : having 

 given the Point of the Curve and the Altitude at the Beginning of the Fall. 

 By John Machiny* Gresham Astron, Profes. and R. S. Seer. N° 358, p. 86o. 

 Translated from the Latin. 



Let the centre of force be c, fig. 6, pi. o ; with which centre and the dis- 

 tance CB, equal to the altitude from which the body falls, let a circle beg be 

 described ; and let bog be a right angle. Let a be the lowest point of the 

 curve, where it meets the axis cb at the given distance ca. It is required to 

 find the point q, where the curve of quickest descent eqa meets the circle qp, 

 at another given distance cp. — This problem has two cases ; one depending on 

 the hyperbola and circle, the other on the ellipse and circle. 



Case 1 . — When the centripetal force is reciprocally as the distance from the 

 centre. Let klm be any rectangular hyperbola, having its centre c and asymp- 

 tote cb, and meeting the perpendiculars bk, fl, am, in the points k, l, m. 

 Make cd to cg as v^aplm to v^abkm, and draw dh perpendicular to cg. Also 

 take the sector rcb to the area hdcb as the given hyperbolic area abkm to the 

 given rectangle ca X am. Then the right line rc will meet the circle fq in 

 the point q, which will be in the curve of swiftest descent eqa. 



And the point b will be found, from whence the fall of the body should be- 

 gin, by taking the sector bce to the quadrantal area bcg, as the hyperbolic 

 area abkm is to the rectangle ca X am. 



Corol. — Hence, if the right line rc, revolved about the centre c, make the 

 sectors rcb proportional to the areas hdcb, in which the squares of the bases 

 CD are taken in arithmetical progression : then the right lines cr will intersect 

 the curve eqa at the distances ca from the centre, which will decrease in geo- 

 metrical progression. 



Case 2. — When the centripetal force is reciprocally as any other power of the 

 distance from the centre. Let n -|- I be the index of that power, n being any 

 number, either integer or fraction, affirmative or negative ; and let h = cb, 

 fig. 7, be the greatest altitude of the required curve eqa, A = ca the least 

 altitude of the same, and a = cp any other intermediate altitude. 



In the right line cg take cd to cb as y^A" to \/h", and also ch to cd as 

 v' a" — A" to ^h" — A". Then with the centre c, and semiaxes cd, cb, de- 

 scribe the ellipse bld, meeting the ordinate hl in l ; and draw the right line 

 LK touching the ellipse in l, and meeting the less axis cd produced in k : then 



♦ Mr. Machin, who was some time professor of astronomy, and secretary of the Royal Society, 

 was elected to the professorship of Gresham College, the l6th of May, 1713, on the resignation of 

 Dr. Torriano, and died June 9, 1751 . His communications to the Society, besides the present paper. 



