38 PROCEEDINGS OP THE AMERICAN ACADEMY 



p being reckoned as usual from 0° to 360° in the direction n. f. s. p. ; 

 the upper sign being used for the preceding, and the lower sign for the 

 following limb. 



Let #1 and t^ be the observed times when the planet, passing north 

 of the centre, has either limb tangent to lines at the angles p and — p 

 from the assumed zero. Then the 1st equation of (1) gives : 



Tj^z=t^-\- — ± \D seep ± \D tan jo sec jo Ap -|- AS sec^jo Ajo | 



(2) 



Tj = #2 + rE ^ ± \D seep =F iZ) tan p sec p A/> -|- AS sec^jo A^? | 



If ^0 be the corresponding time for a third line drawn through the 

 intersection of the other two and bisecting the angle between them, we 

 shall have p = 0, and 



But we have T^ — ^o = ^o — ^i- Hence, putting t =: ^^ ~l~ 'i — ^ <o, 

 and noting that (1 — cos p) see p = tan p tan hp, 



(4) :f ^D tan p tan ^ p — AS tan^ p Ap = -^- t cos 8 



In a similar way, when the planet passes south of centre, we get 



(5) T ^D tan p tan ^ p — A'S tan^ p Ap = y- t' cos 8 

 The addition of (4) and (5) gives, 



(6) i) = iF cot i p [ -V- cos 8 (t + t') cot jo -[- ( as + A'S) tan p Ap]; 

 and their difference, 



(7) A;, = V-cosS(r'-r)^|^^ 



But we have also, putting r = t^ — t^, and r' = t\ — t\, 



,„. AS = J/- r cos 8 cot p — \D Ap see p 



A'S = — J^- /*' cos S cot jo — \D Ap sec p 



whence 



AS -f- ^'S = V- (^ — ^0 ^^'^ ^ ^°* P (nearly) 

 AS — A'S = -V- \r + ^0 ^^^ ^ ^^^P 



which substituted in (6) and (7) give finally, 



(9) D=^ Y- cos 8 cot ;, cot Jp [r + r' — (t — r') ^^] 



1> — T 



(10) Ap = -rj-z,eoip 



+ 



