4 THE ORBIT OF NEPTUNE. 



Theory of Walker. Theory of Kowalski. 



^Tt — 4° ir " — 4° 12' " 



Se _ 52 _ 51 



^e — 3 6 _ 2 53 



8)1 — 8.4 — 8.5 



Thus, it seems that the theory of Kowalski is, on the whole, no nearer the truth 

 than that of Walker, although it was founded on observations up to 1853, when 

 the planet had moved through an arc of sixteen degrees since its optical discovery.='= 

 The cause of this failure to derive a more accurate result is an accidental mistake 

 in the computation of the perturbations of the radius vector by Jupiter, as I have 

 more fully pointed out in the Monthly Notices of the Royal Astronomical Society 

 for December, 1864. 



§ 4. The form which Professor Kowalski finds his equations of condition to 

 assume is illustrative of an interesting and important principle of the method of 

 least squares. By the comparison of his provisional theory with observations, 

 forty-four equations of condition are obtained for the corrections of the four 

 elements 7t, e, s, and n. It is then inquired whether it is possible to determine 

 the orbit of Neptune from the modern observations alone, omitting that of La- 

 lande, the planet having moved through an arc of sixteen degrees. Treating 

 the equations derived from the modern observations alone by the method of least 

 squares, four normal equations are obtained. Two of these equations are, omitting 

 the terms involving the correction of the mass of Uranus, which we do not need, 



— 10.4994 S)i — 21.2661 Ss + 13.0088 e^n + 40.2211 8e = — 324".65, 

 26.9661 8n. — 73.2702 8e + 40.2211 eSn + 139.9967 Se=z — 886 .63, 



and the other two can be transformed into the following : 



— 10.4994 8n — 21.2,661 §s + 13.0073 eSn + 40.2219 8e = — 324.50, 



— 26.9661 Sa — 73.2702 he + 40.2219 ehn + 140.0009 he- — 886.77. 



It will be seen that the last two equations are very nearly identical with the 

 first two. Hence it is concluded that the modern observations alone give only 

 two independent relations between the four unknown quantities sought, and do 

 not suffice, therefore, to determine the elements of Neptune. 



Now, the identity in question does not prove that the modern observations ai^e 

 insufficient to determine the elements, because it is the necessary result of the mode 

 of treating equations of the land in question hy the method of least squares. This 

 can be most easily shown by a theorem in determinants. By the elementary 

 principle of determinants, if we have a number of linear equations between the 

 same number of unknown quantities, of the form 



* The differences of the tiro values of Sir and 6e, which are so small, do not correctly represent the absolute 

 differences of the two theories, owing to the great difference of longitude of perihelion in the two theories 

 proceeding from the different forms given to the perturbations. The real difference Kowalski — Walker is 

 given by the equations 



<5.esin7r=+ 1", 

 6.e cos TT ^ — 13. 



