Febeuaby 27, 1903.] 



SCIENCE. 



331 



Notes on the Short Method of Determining 

 Orbits from Three Observations: A. 0. 

 Leusciiner. 



In order further to simplify tlie compu- 

 tation of preliminary orbits, the author 

 proposes several modifications in the appli- 

 cation of his 'Short Method, etc' {Publica- 

 tions L. 0., Vol. VII., part 1) : 



1. The accuracy of (\ is increased by 

 eliminating the parallax from the second 

 observation through simple corrections ap- 

 plied to the corresponding solar coordi- 

 nates in group I. 



2. When the parallax factors for the 

 three observations differ materially, the 

 accuracy of the geocentric velocities is 

 increased by applying the parallax corre- 

 sponding to the unit of distance to all three 

 observations in the formulae of group II. 



3. No correction for parallax is to be 

 applied to the middle observation on the 

 basis of the successive approximations for 

 (0(1 (cf. groups v., VII.), but instead the 

 parallax is to be eliminated once for all 

 by correcting the rectangular equatorial 

 solar coordinates for the normal date as 

 follows : 



AX= ( paPa) sin Og cos <\ + (psPo) cos % sin <\, 

 ^Y=^ — (PaPo) COS Qo COS (!o + (jJiiPo) sin a^ sin <!„, 

 ^2" = — [piPo) cost'o- 



4. By replacing the A and B by 



A' = A-\-Qo%i palp\ B' = B+ psli>'. 



respectively, in the differential formuliE 

 (group VII.), terms depending on the 

 parallax factors are introduced which will 

 minimize the effect, on the residuals, of 

 changes in parallax, and the convergence is 

 increased. 



5. The sufficiency of the differential 

 formulas should be tested by checldng the 

 new residuals obtained in group VII. by 

 means of the corresponding formuliE of 

 group VI. 



6. Simple formulae involving the squares 

 of the corrections have been derived for 

 those rare cases in which the linear rela- 

 tions ai-e found to be insufficient. 



7. The new values of Xq, y^, z^ (group 

 VII. ) may be found rigidly in all cases by 

 changing the former values by 



?x„ = cos «„ cos (SqSpo , di/^ = sin a^ cos fS„3pD , 

 feo = 8in VPo- 



S. The method may be applied to longer 

 arcs by using closed expressions in place 

 of the series in group VI. 



A Method of Computing Orbits in Rectan- 

 gular Coordinates: A. 0. Leuschner. 

 Prom 



{Publications L. 0., Vol. VII., part 1) the 

 author derives the three fundamental equa- 

 tions : 



„ ^ .9iii ^ ffi ^ 



fi9ni—fm9i ' fiffni~fuil/i '"' 



Introducing 



p cos a cos (5 = X + (X) 



in the first of these three equations, it be- 

 comes 



fiffni — /iiiffi 



ffi 



fiffm — fiuSi 



Pi cos «! cos (?! 



Pii, cos Ojjj cos (^,[1 — Pd cos Oq cos (5|) 



+ (^)o - <?iii(^)i + <?i(X)„i = 



where the (X) (similarly the (Y) and (Z) 

 in the remaining two equations) are the 

 solar coordinates, corrected, to eliminate 

 parallax, by the formulae given in 3 of 

 the foregoing 'Notes.' These coordinates 

 are referred to the beginning of the year 

 and apply to the actually observed dates. 



The solution of the fundamental equa- 

 tions gives at once 



CxZ)' + eyD" 4- '■;,D'" 



