HISTORY OF PHYSICAL ASTRONOMY.
203
the true elements, still, when the question refers only to a section of theorbit, this object may be accomplished by employing indefinite combina-tions of elements, differing very considerably from each other. This willbe readily understood when it is borne in mind that the theory of ellipticmotion assigns four arbitrary constants, which may be modified in a varietyof ways, so as to answer the purpose of mutual correction; and that onaccount of the smallness of the arc described by the body, the outstanding! errors, which inevitably exist in all such cases, are not allowed time todevelope themselves to any serious extent. In the case of Neptune dis-turbing Uranus , the perturbations are sensible only a little before andafter conjunction. Throughout the whole period embraced between 1090,the year of the earliest observation of Uranus , and the commencement ofthe present century, the action of the disturbing planet has been quiteinappreciable; and consequently the tabular errors of Uranus for thatperiod may be considered as wholly explicable by the errors of the ellipticelements. The last conjunction of the two planets took place in the year1822, and the action of the disturbing planet was sensible only during| about twenty years anterior to that event, and the same number of yearssubsequent to it. It is manifest, therefore, that any elements which willafford a pretty accurate representation of the heliocentric co-ordinates ofNeptune during the present century, will account with sufficient fidelityfor the perturbations of Uranus during the same period. Conversely, ifthe perturbations are faithfully accounted for, we may conclude that thetheory is capable of representing the co-ordinates of the disturbing planetwith considerable precision, and that they may be employed with confidencefor the purpose of its actual discovery. That the theories of Le Verrier and Adams were capable of so representing the co-ordinates of Neptune during the whole period when its action was sensible, may be seen fromthe following table, which exhibits the actual and theoretical values of thelongitude and radius vector for the beginning of each of the years specifiedbetween the years 1800 and 1860 :■—
Planet Neptune.
Theory of
Theory of Adams *.
1st Approximation.
2nd Approximation.
Year.
Longitude.
Rad. Vec.
Longitude.
Rad. Vec.
Longitude.
Rad. Vec.
Longitude.
Rad. Vec.
1800
226 “ 4'
30.30
231 ° 34 '
33.57
236 ° 46 '
36.31
238 “ 9 '
34.90
1810
247 20
30.28
251 10
32.80
254 13
34.75
256 39
33.92
1820
268 52
30.23
271 28
32.35
273 11
33.45
276 5
33.25
1830
290 31
30.55
292 8
32.29
293 27
32.56
295 54
32.96
1840
312 17
30.06
312 36
32.63
314 30
32.22
316 10
33.11
1850
334 12
29.96
332 25
33.32
335 36
32.48
335 50
33.67
1860
356 14
29.87
351 17
34.26
356 1
33.30
354 39
34.57
* Mr. Adams’ theory is founded upon a method of successive approximation. A cer-tain value of the mean distance is first assumed, and by a comparison of the resullscalculated by its aid with those derived from observation, an indication is obtained of thedirection in which the error in the mean distance lies, and also of its probable magnitude.The solution is then repeated with a new value of the mean distance, suggested by theoriginal solution, and a further approximation is obtained. This is the method of