7'2 PHILOSOPHICAL TRANSACTIONS. [aNNO I763. 



of the sun to the tangent of the orbit of the earth, at the point where the earth 

 is, to a line drawn in the same angle from the sun to the tangent of the orbit of 

 the planet projected on the plane of the ecliptic, at the place of the planet in the 

 ecliptic. 



Let A be the sun, bc the orbit of any planet, de the same projected on the 

 plane of the ecliptic, fg being the line of the nodes, b the place of the planet in 

 its orbit, d its projected place : then the plane through b and d, which shall be 

 perpendicular to both the planes bc and de, intersecting those planes in bh, dh, 

 the lines bh, dh will be both perpendicular to the line of the nodes, and the 

 angle bhd the inclination of the orbit to the plane of the ecliptic. But tangents 

 drawn to bc and de, at the points b and d respectively, will meet the line of the 

 nodes, and each other in the same point i, and the velocity of the planet in lon- 

 gitude will be to its velocity in the orbit bc, as di to bi. 



Now from the point a let ak fall perpendicular on bi, and al be perpendicu- 

 lar to Di: then the ratio of di to ib will be compounded of the ratio of di to dh, 

 or of ai to AL, of the ratio of dh to bh, and of that of bh to bi, that is of ak 

 to AI. But DH is to BH as the cosine of the inclination of the orbit to the ra- 

 dius; and the two ratios, that of ai to al, and that of ak to ai, compound the 

 ratio of ak to al: therefore the velocity of the planet in longitude, is to the ve- 

 locity in its orbit, in the ratio compounded of that of the cosine of the inclination 

 of the planet's orbit to the radius, and that of ak to al. 



Moreover the ratio of the velocity of the planet in b to the velocity of the 

 earth in any point of its orbit, is compounded of the subduplicate of the ratio 

 of thelatus rectum of the greater axis of the. planet's orbit to the latus rectum of 

 the greater axis of the earth's orbit, and of the ratio of the perpendicular let fall 

 from the sun on the tangent of the earth's orbit at the earth to ak, the perpen- 

 dicular let fall on the tangent of the planet's orbit at b. Therefore the velocity 

 of the planet in longitude, when in b, to the velocity of the earth in any point 

 of its orbit, is compounded of the subduplicate ratio of the latus rectum of the 

 greater axis of the planet's orbit, to the latus rectum of the greater axis of the 

 earth's orbit, of the ratio of the cosine of the inclination of the planet's orbit to 

 the radius, and of the ratio of the foresaid perpendicular on the tangent of the 

 earth's orbit to al, the perpendicular on di : these perpendiculars being in the 

 same ratio with any lines drawn in equal angles to the respective tangents. 



This being premised, the place of a planet in the ecliptic being given, the 

 place of the earth, whence the planet would appear stationary in longitude, may 

 be assigned thus, a denoting the sun, fig. 23, let b be a given place of any 

 planet in its orbit projected orthographically on the plane of the ecliptic, cb the 

 tangent to the planet's projected orbit at the point b, which will therefore be 

 given in position. Also let de be the orbit of the earth, and the point d the 



