VOL. LIII.] PHILOSOPHICAL TRANSACTIONS. 73 



place of the earth, whence the planet would appear stationary in longitude at b. 

 Join AB, and draw a tangent to the earth's orbit at the point d, which may meet 

 CB in F, and the line ab in o ; draw also ah making with df the angle ahd equal 

 to that under abc. Then the point d being the place whence the planet appears 

 . stationary in longitude, as fb to fd so will the velocity of the planet in longitude 

 in B, be to the velocity of the earth in d ; this velocity of the planet in b being 

 also to the velocity of the earth in d, in the ratio compounded of the subdupli- 

 cate of the ratio of the latus rectum of the greater axis of the planet's orbit to 

 the latus rectum of the greater axis of the orbit of the earth, of the ratio of the 

 co-sine of the inclination of the planet's orbit to the plane of the ecliptic to the 

 radius, and of the ratio of ah to ab ; therefore the ratio of fb to fd will be com- 

 pounded of the same ratios ; and if r be taken so that the ratio of ab to i be 

 compounded of the first two of these, i will be given in magnitude, and the ratio 

 of FB to FD will be compounded of the ratio of ab to i, and of ah to ab. Whence 

 FB will be to FD as ah to i ; and the angles cba, or fbg, and ahg being equal, by 

 which FG will be to f^ as ag to ah, by equality fg will be to fd as ag to i, and 

 DK being drawn parallel to fb, bg will be to bk as pg to fd, and therefore as 



AG to I. 



But now as this problem may be distributed into various cases, in the first place 

 consider the earth as moving in a circle concentric to the sun, and likewise cb, 

 the tangent to the planet's orbit, perpendicular to ab. But here dk also will be 

 perpendicular to ab, and ab meeting the earth's orbit in l and m, the rectangle 

 under kag will be equal to the square of am, fig. 24. But bg being to bk as ag 

 to I, if BN be taken equal to i, bg will be to bk as ag to bn, and ab to kn also as 

 AG to bn, and the rectangle under nk, ag equal to that under ab and i ; there- 

 fore the rectangle under kag being equal to the square of am, nk will be to ka 

 as the rectangle under ab and i to the square of am, that is, in a given ratio, and 

 kd with the point d will be given in position. 



Again, when cb is not perpendicular to lm, let do be perpendicular to lm. 

 Then the rectangle under oag will be equal to the square of am, fig. 25. But 

 bn being taken equal to i as before, the rectangle under nk, ag will be equal to 

 that under ab, i, whence nk will be to ao in the given ratio of the rectangle 

 under ab, i to the square of am. Therefore np being taken to pa in that ratio, 

 the point p will be given, and kp, the excess of np above nk, will be to po, the 

 excess of ap above ao, in the same ratio. Hence, as dk is parallel to cb, and do 

 perpendicular to lm, the triangle kod is given in species ; and if pd be drawn, 

 the angle opd will be given ; for the co-tangent of the angle okd will be the co- 

 tangent of the angle opd, as ko to op, that is, as the rectangle under ab, i to- 

 gether with the square of am, to the square of am : and hence the point d is 

 given by the line pd drawn from a given jx)int p in a given angle apd ; and if ad 



VOL. XII. L 



