VOL. LVIIl.] PHILOSOPHICAL TRANSACTIONS, 523 



(ill), the planet's altitude, being = c + e ; where the sign — is used in the 

 former, and ■+■ in the latter case of the rule; excepting in the former case those 

 places of the earth's disk more south than the sun's centre, and in the latter the 

 places more south than the same-centre, provided f + the com[)lement of q ^ 

 90°; and this exception will also hold when the planet's path, at either node, is 

 more south than the sun's centre. But if the planet transit the sun with north 

 latitude, the exception in respect of the places is to be inverted, so that the sign 

 + takes place in the former case of the rules, when the place of the earth's disk, 

 more north than the sun's centre, gives p + compl. a >■ 90°; but that being 

 the same, in the latter cases of the rule, in regard to terrestrial points more 

 south than the sun's centre, the sign — takes place. Now put the horizontal 

 parallax of the planet from the sun =: h, which in investigating the parallax is 

 assumed at pleasure, but yet so as to be not far from the truth ; put also the 

 parallax in altitude of the planet from the sun cl = p; then will p= H. cos 

 (c ^ e); or, if E be neglected, p := h. cos c. 



§ 4. These being premised, we are now to give formula for computing the 

 effects of parallax about the moments or times of the contacts. Let then k ^S, 

 fig. 10, be the ecliptic at the descending node, or k g^, fig. 1 ), at the ascending 

 node ; oe the apparent path of the planet as seen from the earth's centre; c the 

 common centre of projection of the earth and sun [§ i and 11] ; mr the celestial 

 meridian; CD the planet's latitude at the moment of the ecliptic conjunction; l 

 any given place in the earth's disk facing the sun ; then it will appear, from what 

 is above, that the angle pcl = a, and cl = p. Now join dl ; and put cd = n; 

 also the angle made by the meridian with the ecliptic, viz. the angle rc ?$ or 

 RcS^ = ^, and the angle lcd = r; and there will arise a double form for the 

 value of r, according as the planet's path, at the former or latter node, shall be 

 more south or north than the sun's centre; viz. if the path be more south at ^, 

 or at Q, more north than the sun's centre, it will be r = Q0° -\- b — a, where, 

 instead of b is to be taken its supplement to 180° when the observation is made 

 before noon : but the planet's path being at 'Q more north, or at Q, more south 

 than the sun's centre, it will be r = « 4I i + 90°, where the upper signs take 

 place in observations made afternoon, but the lower signs before noon. Further, 



make 90" — xr = / ; and '■^-^^^~. tang, t = tang, x; also put the angle cdl = 1/ , 



then will y ■=. t +_ x, where the sign — takes place when w >• p ; hence dl 



=: —. . Since there is riven the angle of the planet's path with the circle of 



smy ° fc r r 



latitude, which call e, so that in fig. 10, the angle edi = e, and in fig. 1 1 the 



angle ode = e ; hence there will also be given the angle ldf or lde. If now 



with centre l, and radius equal to the sum or difference of the semidiameters of 



the sun and planets, which call m, the path be cut in i and e ; then, because of 



3x2 



