June 8, 1917] 



SCIENCE 



573 



tions. Since the choice of these planes is 

 arbitrary, as long as their intersection coin- 

 cides with the line of sight Olbers reduces 

 the number of available conditions by re- 

 jecting one of the arbitrary planes for the 

 middle place or second observation and 

 adopts for the other arbitrary plane that 

 which corresponds to a great circle drawn 

 through the observed place of the body and 

 through the sun. 



Since the three distances of the body are 

 not furnished by observation they enter the 

 problem as additional unknowns. Usually 

 the distances are derived first, whereupon 

 the solution of the elements becomes com- 

 paratively simple. In Olbers 's method one 

 of the fundamental relations for the de- 

 termination of the distances at the first 

 and third dates has the form pjjj=Mpi, 

 where M is equal to the product of the 

 ratio of two triangular areas into the 

 ratio of the trigonometrical sines of the 

 perpendicular arcs drawn from the first 

 and third observed places, respectively, to 

 the great circle through the sun and the 

 second observed place. The ratios of the 

 triangles referred to form a very important 

 consideration in many orbit methods. The . 

 triangles are contained between successive 

 radii vectores from the sun to the body. 

 For short arcs or intervals these triangles 

 differ but little from the corresponding 

 sectors bounded by the conic, and since 

 according to the law of areas the sectors 

 are proportional to the intervals, the tri- 

 angles are very nearly proportional to 

 the intervals. The ratios of the triangles 

 may then be developed in series of which 

 the first term is identical with the ratio of 

 the intervals and of which the later terms 

 contain the powers and products of the 

 intervals, the inverse powers of the helio- 

 centric distances r and their derivatives. 

 They may be made to depend on the second 

 heliocentric distance r and its derivatives. 



Since r and its derivatives depend on the 

 elements in the orbit their values in gen- 

 eral can not be known until the first ap- 

 proximation has been accomplished by 

 placing the ratios of the triangles equal to 

 the ratio of the intervals. The series repre- 

 senting the ratios of the triangles have 

 been the subject of intensive study in con- 

 nection with the orbit methods resting on 

 the integrals of Newton. The most exhaus- 

 tive study of the true radii of convergence 

 of series of this type is due to Moulton. 

 He demonstrates analytically the empirical 

 conclusions of astronomers that the series 

 may lose their applicability for comets ob- 

 served near perihelion at a moderate dis- 

 tance from the sun, while for minor planets 

 in general they give universal satisfaction. 

 In referring to the indeterminateness in 

 Olbers 's method I am not at this moment 

 concerned with any inaccuracies that may 

 arise from his using in the first approxi- 

 mation the ratios of the intervals for the 

 ratios of triangles. The indeterminate- 

 ness I am referx'ing to arises from the fact 

 that when the first and third observed posi- 

 tions lie on the auxiliary great circle 

 through the second place and the sun, re- 

 ferred to above as being introduced by 

 Olbers, then both the perpendiculars from 

 the first and third places on this great circle 

 become zero and M becomes indeterminate. 

 It becomes nearly indeterminate when the 

 three observations lie approximately in the 

 great circle through the sun, and the degree 

 of indeteiTninateness in such cases depends 

 upon the magnitude of the errors of ob- 

 servation as compared with the magnitude 

 of the perpendiculars. It is evident that 

 perpendiculars of but a few seconds accu- 

 rately derived from precise observations 

 would still yield a working first approxi- 

 mation, while larger perpendiculars com- 

 parable to the errors of observation would 

 lead to fallacies or yield nothing. Here 



