202 PROCEEDINGS OF THE AMERICAN ACADEMY 



If a perpendicular be let fall from the central body on the straight 

 line which joins the earth and the body whose orbit is to be deter- 

 mined, its length is obviously 



Ji sin (X — L) ; 



another expression for the length of the same line is 



rs i n ( x — *X). 



Hence for the three times of observation, the three equations 



r sin (x_i — X_0 = #_i sin (X_ 1 —L_ 1 ), 

 r sin (xo — *o) = ^o sin (X — L ), 

 r sin (xi — X : ) = i? x sin (X x — L x ). 



But since the orbit is circular, x increases uniformly with the time, 

 and consequently xo — X-i = Xi — Xo = V suppose. 

 Thus the above equations may be written 



r sin (xo — n — X_j) = I2_ 1 sin (X_ x — Z_j) = a_ v 

 r sin ( X o — X ) = ^o sin (X — L ) = a , 

 r sin (x + rj — X x ) = R x sin (X x — L x ) = a x , 



which serve to determine the three unknown quantities r, xo> and 77 ; 

 and it will be noticed that their right-hand members are known quan- 

 tities. 



If the sum of the masses of the central body and the body whose 

 orbit is sought is denoted by /z, and the common interval of time be- 

 tween the observations by t, 



thus, if ft were known, two observations would suffice to determine the 

 orbit ; but if fi is not known, 77 must be regarded as an independent 

 unknown quantity. Hence the necessity for the restriction put at the 

 end of the statement of the problem. Also by this restriction the 

 problem is made to depend on the solution of an algebraical equation 

 instead of a transcendental one. 



The equations can be simplified by taking two unknown quantities, 

 <•> and a, instead of xo ar >d q, such that 



Xx + X_! 



= Xo — 



2 

 Xi-X_i 



