140 Astronomical and Nautical Collections. 



labour. We might therefore be contented to determine either 

 the longitude of the node and the inclination of the orbit, or the 

 time and distance of the perihelium, according to the mode of 

 correction which we might prefer. Thus if ;)(; be = m"' — u', or 

 the angle formed by the two positions of the revolving radius, 



we have cos v = 7r-,-r, > and thence ip by the formula 



2rr ^ *' 



r 1 



tang i (p=:cot I X — V -V,' -■ — ^ — ' which enables us to de- 

 r sin ^ X 



termine the time and distance. We may also find tp more 



A- .1 • -21 k"-(r"'-r'y . . (r"' + ry—k"^ 



airectly,smcesm'i v=: ^— ; — - and cos-|y=:- _: — — 



4 r r 4r r' . 



fr"' + r'y k"^ 



whence cot' h x— —rjz —-, ;tj » consequently tang ^ <p = 



^([r"' + r'f-k"')-2r' 

 r [k"'' — (r'" — r'y 



§44. 



It will be convenient to recapitulate the formulae immedi- 

 ately necessary for the computation, that the whole may be 



tano- B" 

 found together. We first find m = - — 2 — then M = 



sm (A' — a ) 



(^sin(A'-.')-tanggOr ^ ^^^ ^^^ coefficients of 

 (tang^' — wisin[A' — a"'])^' 

 r'" =: R'- — 2 R' cos (A' — «') §' + sec^ (3'/^ 

 r'"" - R"'= — 2 R cos (A'" — a.'") M/ + sec" 0" M-/' 

 k'" :^r" + r"" - 2 R' R'" cos (A"' - A') + 2R'" cos (A'" -a') §' 

 + 2M R'' cos (A' —a.'")/— 2 Mcos («'"-«') §'»— 2M tang(3, 

 tang 0" ^"^ ; hence we obtain ^ by a few trials, and from it 

 all the other elements of the orbit, by means of the formulas in 

 §. 42 and 43. 



§ 45. 



Even a superficial comparison of this method with any other 

 that has been hitherto proposed, will be sufficient to show its 

 superior conciseness and convenience. It has also the ad- 

 vantage of being universally applicable, whenever we have three 



