determining the Orbits of Comets. 17 



it is clear that g expresses the chord of the earth's orbit 

 between E and E", and G the longitude of the first place of 

 the earth when seen from the third. Combining in the same 

 way the second system with the third, and assuming 



x" — Xj ass M p cos ot" — p cos a = ph cos £ cos H 



(20) y" — y / = M p sin a" — p sin a = p h cos ? sin H 



z"-is / sM/) tan 8" — p tan 8 = p h sin ?, 



p # will be the distance of the point N from C", and £ and H 

 are the latitude and longitude of C" when seen from N. Asp 

 disappears in the equations, ft, £, and H are known quantities. 

 The systems (19) and (20) combined together give, 



x" — x = x" — x / — (x — xj), &c. 



(21) A 9 = p*#-2gpAcos?cos(G- H) + g 2 . 



The formulae (18) and (21) assume a form more convenient 

 for logarithmic calculation by representing them as sums of 

 two squares. 



With this view let 



cos 8 cos (a — 6>) = cos 4/ 



(22) cos 8" cos (a" — 0") = cos ^' 

 cos ? cos (G — H) = cos $| 



and we have 



r = */((? sec 8 - R cos vj/) 2 + R 9 sin \J/ 2 ) 



(23) r"= */((M p sec 8" - R" cos <}/') 3 + R" 2 sin ^" 2 ) 



k = \/((p h -s cos $) 2 + S 2 sin ♦*) • 



Here ty and \(/" are the angular distances of the comet from the 

 sun, and <p is the angle at N in the triangle N C C". Lastly, 

 the calculation may be still more simplified by introducing a 

 new variable quantity. If we suppose 



ph — g cos $ = u, or 

 __ u + g cos £ 



and substitute this value in the first two equations of (23), 

 every quantity will be expressed by u. As g sin $ is the per- 

 pendicular line drawn in the triangle N C C" from C to the 

 side N C /r , so u is the distance from C ff of the point of inter- 

 section of this perpendicular line with the line N C /r . 



The formulae here given, which, it appears, have the form 

 most convenient for calculation, have been published by Gauss 

 on the occasion of the second comet of 1813*. In the calcu- 



* Zach's [German] Monthly Correspondence, vol. xxviii. p. 504. 

 Third Series, Vol. 7. No. 37. July 1835. D 



