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 corresponding values from Gauss[1] and Oppolzer[2] 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.
| First hypothesis | .4282377 | .4132937 | .4061399 | ||||
| Second hypothesis | .4282782 | .4132809 | .4061998 | ||||
| Third hypothesis | .4282786 | .4132808 | .4062003 | ||||
| Gauss: | |||||||
| First hypothesis | .4323934 | .4114726 | .4064712 | ||||
| Second hypothesis | .4291773 | .4129371 | .4071975 | ||||
| Third hypothesis | .4284841 | .4132107 | .4064697 | ||||
| Fourth hypothesis | .4282792 | .4132817 | .4062033 | ||||
| Oppolzer: | |||||||
| First hypothesis | .4281340 | .413330 | .4061699 | ||||
| Second hypothesis | .4282794 | .4132801 | .4061976 | ||||
| Third hypothesis | .4282787 | .4062009 | |||||
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 corresponding determination by the method of the present paper requires the successive corrections of three independent variables.
