Sg6 PHILOSOPHICAL TRANSACTIONS. [aNNO I7I9. 



Let ABP, %. 14, pi. 10, be the elliptical orbit of a planet, ap the transverse 

 axis, CB the semi conjugate, s the sun, a the other focus of the ellipse. Through 

 s draw sm parallel to cb : then m will be the point where the sun's distance 

 increases or decreases the quickest, and sm = ac — — . 



AC 



And if SL be taken a mean proportional between the semi-axes ac, cb, 

 then L will be the place of the greatest equation of the centre, as they call it ; 

 or where the angular motion is equal to the mean motion. If the excentricity 

 be not greater than what it is in most of the planets, then bl = ^bm nearly: 

 and sl := v^ ^ ac* — ac^ sc". 



If there be required the point n, in which the real motion in the curve 

 changes the quickest, the problem is a solid one. For 2ns = 4ac — 2Na is 

 to 3ng — AC as Ac"^ — cs^ = CB^ is to Na^: and therefore, putting ac = a, 

 CB = c, and nq = ?/, we shall have the equation y^ — 2ai/y + 4cc^ — -^cc = 0; 

 which being resolved, it will give y or ng the distance of the required point n 

 from the other focus. But in orbits that are but little excentric, as those of 

 the planets, if there be made cd =: sq, and ak = ad, then the remaining 

 part of the axis pk will be = ns the sun's distance from n very nearly. But 

 if the orbit be parabolical, sn will be to sp as 5 to 4, and the angle nsp = 

 53° 8' nearly, its sine being ^ of the radius. 



But the point o, in which is the greatest acceleration of the apparent or 

 angular motion of the descending planet, or the greatest retardation of the 

 ascending, will be obtained in this manner: In ac take cg =: ^ac, and make 

 csp an angle of 30 degrees, draw sf, and take ce equal to it, also take gh = 

 ge: then if so be made = ph, the point o will be the place of greatest change 

 of the angular motion of the planet, revolving in the elliptical orbit abop: for 

 in that place of the orbit, the second differences of the equations of the centre 

 of the planet will be found the greatest: and so = -g-AC — -^ -j^ac^ -j- ^sa'^ 

 But when the orbit is parabolical, as in the comets, take so to sp as 8 to 7, 

 then the angle osp will be 41° 14%, or its sine is to radius as ^^7 to 1. 



Lastly, the direction of the tangent of the orbit will change with the least 

 velocity in the point e, if sr be taken = ^ab. If the excentricity sc be less 

 than -^pc, this minimum does not take place, but this velocity with which the 

 tangent revolves is always decreasing, as far as to the aphelion ; as it is in the 

 motions of all the planets. Neither does it obtain in a parabolic orbit, because 

 of its axis being produced in infinitum. 



All these things are demonstrated from the foregoing theorems of Mr. 

 Demoivre, by the precepts in tlie doctrine of maxima and minima. 



