12 Prof. Encke on Olbers's Method of 



annexed table for all values of >j, from >j = to rj = \ V 8, in- 

 creasing by hundredth parts of unity. The logarithms by 

 which these calculations were made being only correct to 7 

 decimals, the last figure may be wrong one or at most two 

 units, which, however, will have the less influence, as Lam- 

 bert's formula is in this way applied with much greater ac- 

 curacy than in the common way. In the latter, as k is small 

 in comparison of r + r 1 , a small quantity is obtained as the 

 difference of two greater ones, while there is no such subtrac- 

 tion in the use of the table. The case oft/ ~u7 180° has not 

 been taken into consideration, as the interpolation between the 

 values of log ft would then be too tedious. Besides, in these 

 rarer cases there is not the same disadvantage in following the 

 common process. 



Lambert's formula shows most clearly how many observa- 

 tions are required for determining the parabolic orbit. If the 

 distance of a comet from the earth were known for one obser- 

 vation, its place in space, consequently likewise its place with re- 

 gard to the sun, would be given. Every observation which is 

 used introduces, therefore, one unknown quantity ; for two ob- 

 servations two radii vectores and the connecting chord k might 

 be expressed by two unknown distances, but only one equation 

 between r, r 1 and k would be derived from them. But three 

 observations would give three equations for three unknown 

 quantities, by combining every two of the three radii vectores; 

 this would be consequently sufficient for solving the problem. 

 This application would satisfy one of Kepler's laws, viz. that 

 the times are proportional to the areas described ; but it is 

 quite independent of the second equally important law, that the 

 comet must move in a plane passing through the sun. This 

 latter law gives for the three places of the comet an equation of 

 condition, so that in three complete observations there are four 

 equations with three unknown quantities. Although, there- 

 fore, two observations are not sufficient for determining the 

 parabola, yet three observations are more than enough, and it 

 will be necessary either not perfectly to satisfy one of the data 

 contained in the three complete observations, or only to fulfill 

 one condition resulting from a combination of two data while 

 the other is entirely neglected. 



Olbers's method is founded on the condition that the three 

 places of the comet shall be in one plane with the sun. It de- 

 termines from it with a close approximation the ratio of two 

 distances from the earth, and consequently also two places of 

 the comet in space, expressed by one unknown quantity. This 

 ratio applied to Lambert's equation between the places and the 

 time would give, if the substitution were actually made, one 



