CHAPTER IV. 



RESULTS OF THE COMPARISON OF THE THEORETICAL WITH 

 THE OBSERVED POSITIONS OF NEPTUNE. 



29. THE first question of the present chapter will be whether the observations 

 of Neptune can be satisfied within the limits of their probable errors by suitable 

 changes in the elements of the orbit of Neptune and the masses of the disturbing 

 planets. 



No admissible change in the mass either of Jupiter or Saturn will sensibly 

 affect the perturbations of Neptune. The mass of Uranus will, therefore, be the 

 only one the correction of which need be taken into account. 



The errors of the provisional latitude of Neptune are so small that the errors 

 of the longitude in orbit may be taken as sensibly the same with the errors of 

 ecliptic longitude. The latter give equations of condition between the following 

 unknown quantities. 



Correction of the mean longitude of Neptune, 

 mean motion of Neptune, 

 eccentricity X sm - perihelion of Neptune. " 

 eccentricity X cos - perihelion of Neptune. 

 " " mass of Uranus. 



But if we attempt to solve by least squares the equations between these cor- 

 rections, we shall be met with the difficulty set forth in the introduction, and our 

 normal equations will be equivalent to only three, unless we include a great 

 number of decimals in the computation. We shall, therefore, make a linear 

 transformation of the unknown quantities, on the principles already referred to, 

 and suggested by the following considerations. 



The true longitude of Neptune has been less than its mean longitude, and its 

 true motion has been greater than its mean motion, ever since its optical discovery. 

 From these circumstances the difficulty in question arises. We may obviate it 

 by substituting for the mean longitude and mean motion of Neptune during an 

 entire revolution its average longitude and heliocentric motion during the period 

 of the modern observations. Suppose an imaginary planet to move uniformly in 

 the orbit of Neptune in such a way that its average longitude and motion have 

 been the same as the average longitude and motion of Neptune during the last 

 nineteen years, and let x be its longitude, 1850, Jan. 0, and xf its annual motion. 

 We may then make the eccentricity and perihelion of Neptune to depend ana- 

 lytically upon the deviation of its motion from that of the hypothetical planet, 

 as it must depend really, because this deviation is the only real datum which we 

 to reason from, the Lalande observations excepted. It is to be remarked 



9 May, 1865. 65 



