148 



DETERMINATION OF ELLIPTIC OEBITS. 



The immediate result of each hypothesis is to give three positions of 

 the planet, from which, with the times, the orbit may be calculated in 

 various ways, and with different results, so far as the positions deviate 

 from the truth on account of the approximate nature of the hypothesis. 

 In some respects, therefore, the correctness of an hypothesis is best 

 shown by the values of the geocentric or heliocentric distances which 

 are derived directly from it. The logarithms of the heliocentric 

 distances are brought together in the following table, and correspond- 

 ing values from Gauss* and Oppolzert are added for comparison. It is 

 worthy of notice that the positions given by our second hypothesis are 

 substantially correct, and if the orbit had been calculated from the 

 first and third of these positions with the interval of time, it would 

 have left little to be desired. 



In comparing the different methods, it should be observed that 

 the determination of the positions in any hypothesis by Gauss's 

 method requires successive corrections of a single independent variable, 

 a corresponding determination by Oppolzer's method requires the 

 successive corrections of two independent variables, while the corre- 

 sponding determination by the method of the present paper requires 

 the successive corrections of three independent variables. 



* Theoria motus, 159. 



t Lehrbuch zur Bahnbestimmung der Kometen und Planeten, 2nd ed, , vol. i, p. 394. 



