DETERMINATION OF ELLIPTIC ORBITS. 



I. 



From the given times we obtain the following values : 



139 



Control : 



+ 2 +^.3 = 2-4959086 

 Tl r 3 = 2-4959081 



II. 



From the given positions we get : 



III. 



The preceding computations furnish the numerical values for the 

 equations III 15 IIF, III 2 , III", III 3 , III'", which follow. Brackets 

 indicate that logarithms have been substituted for numbers. 



We have now to assume some values for the heliocentric distances 

 r i> r 2> r s- A mean proportional between the mean distances of Mars 

 and Jupiter from the Sun suggests itself as a reasonable assumption. 

 In order, however, to test the convergence of the computations, when 

 the assumptions are not happy, we will make the much less probable 

 assumption (actually much farther from the truth) that the helio- 

 centric distances are an arithmetical mean between the distances of 

 Mars and Jupiter. This gives '526 for the logarithm of each of the 

 distances r lt r- 2 , r 3 . From these assumed values we compute the first 

 column of numbers in the three following tables : 



