THE OKBIT OF NEPTUNE. 73 



The coefficient of 8v will be sensibly unity, and that of fy will vary from about 

 0.5 to + 1.0 in each series. 



Error of theory by tJie Lalande observations. 



+ 2".3 



(It will be remembered that the probable error of the Lalande position was 

 estimated at 2".8 ; but, owing to the over-estimate of the comparative precision of 

 the modern observations, the weight assigned to this position in the equations of 

 condition corresponded to a probable error of rather more than 4".) 



By modern observations. 



Limiting dates. Error of longitude. Error of parallax. 



1846-47, O."o5 o"l8 



1848-50, -0.08 0.03 



1851-53, 0.07 +0.55 



1854-56, 0.08 0.00 



1857-59, + 0.22 + 0.23 



1860-62, + 0.11 + 0.18 



1863-64, + 0.02 + 0.28 



These errors are as small as could be expected if the theory were perfect. 

 There is, therefore, no indication of the action of an extra-Neptunian planet. 

 But this fact does not militate against the existence of such a planet. The per- 

 turbations of a planet, and its elliptic elements, develop themselves, not in pro- 

 portion to the time, but in proportion to the square of the arc described. .In 

 order, therefore, to determine the errors of a slow-moving planet with as much 

 accuracy as those of a quick-moving one, we must observe it through a period pro- 

 portioned to its time of revolution. And we cannot detect a deviation of long period 

 from an elliptic orbit until we have accumulated data much more than sufficient 

 for the exact determination of the elliptic elements. For example, when the 

 position of Neptune was determined from the perturbations of Uranus, the latter 

 planet had been regularly observed through an arc of some 270. Moreover, the 

 two planets had been in conjunction in 1824. They are also remarkably near 

 each other when in conjunction. Yet, with all these circumstances so favorable 

 to the development of large perturbations, Uranus only wandered about 5" from 

 an elliptic orbit during the entire period of the modern observations. 



Perturbations will, at first, be developed in proportion to the square of the arc 

 passed over. Therefore, had Uranus been observed through an arc of only 120, 

 the perturbations by Neptune would have been indicated only by deviations in 

 heliocentric longitude of less than I". It is, therefore, almost vain to hope for the 

 detection of an extra-Neptunian planet from the motions of Neptune before the 

 close of the present century. 



34. Determination of the position of the plane of the orbit of Neptune. 



To determine the corrections of the constants p and q, which determine the 



10 Hay, 1865. 



