208 PROCEEDINGS OF THE AMERICAN ACADEMY 



± 1, it indicates that the two corresponding values of tan o- are imagi- 

 nary. The ambiguity in the determination of o- from its tangent is to 

 be removed by taking it in that quadrant which permits the equation 



sin 2 a = 2 c sin (o- -\- /3) 

 to be satisfied. 



Although all these roots will satisfy the equations with which we 



began this discussion, yet they do not all necessarily belong to the 



problem. The reason of this is, that the three equations are not a 



complete statement of all the conditions of the problem. If we denote 



by A the distance of the body, whose orbit we are determining, from 



the earth, we shall have 



A_j = r cos ( X o — v — A_i) + -R-i cos (X_ x — L_{), 

 A = r cos (xo — Xo) + ^o cos (X — L ), 



A x =r cos ( X o + 7 — *i) + -#i cos ( x i — A)- 

 The conditions of the problem demand that A_ x , A and A x shall be 

 essentially positive. Hence, if any system of values of r, xo and rj ren- 

 ders any of these quantities negative, it must be rejected. These re- 

 jected solutions really belong to the problem when one or more of the 

 quantities X_ x , X and X 2 are increased by 180°. In fact, on referring 

 to the equations with which we started, we see they are not altered 

 when any one of the quantities X is increased by 180°. The geometri- 

 cal statement of the problem is more comprehensive than the applica- 

 tion of it to the discovery of the elements of circular orbits. Instead 

 of the above criteria for the rejection of solutions not applicable, the 

 following, which is simpler, may be used, viz. that x always must lie 

 in the angle between L -\- 180° and X which is less than 180°. 

 This example is added for the sake of illustration : — 

 Suppose in the case of Venus revolving about the sun we have these 

 data, 



Wash. Mean Time. A L log R 



1869 Jan. 1.0 250° 22' 59".l 281° 24' 54".9 9.9926528 

 " June 15.0 94 37 54. 9 84 33 34. 1 0.0069342 



" Nov. 27.0 292 3 21. 2 245 32 49. 3 9.9939666 



There will be found 



log a_! = 9.7048977 n , log a = 9.2497072, log a y — 9.8545925, 

 log* =0.5426896, £ =324° 41' 4."52, 8 = 176°35' 15."25, 

 loga =9.7678074 n , log b =9.3111404. 



