Royal Astronomical Society. 71 



A Method of Calculating the Orbit of a Planet or Comet from 

 Three observed Places. By Professor Challis. 



This method resembles in principle that of Laplace, of which it 

 may be regarded an extension, the object of the author being to in- 

 clude in the calculation differential coefficients of thethird andfourth 

 orders, for the purpose of ensuring greater accuracy in the final results. 

 The equations by which the problem is solved are formed as follows. 

 If a and |6 be the observed right ascension and north polar distance 

 of the body at one of the given times, corrected to a given equinox 

 and given position of the earth's equator, and x, y, z be its co-ordi- 

 nates at the same time, having their origin at the place of the ob- 

 server, these quantities are related to each other by the two equations 



<r=y cot a, x=z cos a tan /3. 



Each observed place furnishes two such equations. To include 

 parallax, the origin of co-ordinates is transferred to the earth's centre. 

 If p be the distance of the body from the earth, and q be the aberra- 

 tion constant, the effect of aberration is taken into account by chan- 

 ging x, y, z respectively into 



dx dy dz 



x -dT qp > y ~dt qp ' z -li qp - 



The origin of co-ordinates is then transferred to the centre of the 

 sun, by calculating exactly the sun's co-ordinates at the three times 

 of observation. Thus six equations are formed in which the unknown 

 quantities are the heliocentric co-ordinates of the body. The co- 

 ordinates at the first and last times of observation are expressed in 

 terms of the co-ordinates at the intermediate time, by series including 

 differential coefficients of the fourth order of the latter co-ordinates. 

 The six unknown quantities to be found are, then, the heliocentric 

 co-ordinates Xo, y 2 , £ 2 at the middle time, and their first differential 



dx dii dz 

 coefficients — 2 , -*-?, — -. A first solution is obtained by inclu- 

 de dt dt J 



ding only differential coefficients of the second order, and neglecting 



the aberration terms. This conducts to the following values of the 



co-ordinates, — 



*. 2 =M+4, y.=M'+£L, z,=M"+™ 

 r 2 3 r 3 3 r 2 3 



r 9 being the body's heliocentric distance. Hence 



r., 



=( M+ £) 8+ ( M ' + ^ + ( M "^)- 



For solving this equation, a graphical method given by I. I. Wa- 

 terston, Esq., in the Monthly Notice of the Royal Astronomical 

 Society for December 1845, is recommended. The value of r 2 being 

 found, those of x^, y 2 , z 2 , and their first differential coefficients, are 

 readily derived, the equations for determining them being linear. 



By means of the first approximate values of the unknown quanti- 

 ties, the second order of approximation is proceeded with so as to 

 include differential coefficients of the third order and the more im- 

 portant aberration terms. The third approximation included diffe- 



