80 TABLES OF NEPTUNE. 



+ 2551".117 sin I — 2403".358 cos 

 + 1.163 sin 2 ? — 18 .580 cos 2 Z 

 ■ — 0.088 sin 3 Z — .104 cos 3 Z 



2. The change in the equation of the centre produced by the perturbations of 

 the elements h and h during that revolution of the planet which commenced 

 1779, Jan. 4, and ends 1943, Oct. 15. This change is represented by 



2Usinl — 2hh(iosl, 



Sh and Sic being taken from the table on p. 39 for the times corresponding to the 

 various values of / during the period in question. 



3. The terms 



— 14" sin I — 14" cos I 

 — 3 sin 21 — 3 cos 2 I 



introduced to destroy the effect of the constants added to the values of i-Ji, P^^, 

 P,,2, and P„2 to render them positive. 



4. The constant 



3529", 



added to render all the numbers of the table positive. 



During the revolution to which Table XV. corresponds, the planet passed from 

 180° mean longitude, and returned to the same point in the heavens; whence the 

 table begins and ends with this value of I. But since the commencement of the 

 table corresponds to the values of h and k in 1779, and the end to these values 

 in 1943, they do not correspond with each other. The sum of the constants 

 added to Tables V. to XV. inclusive is 1°, which has been subtracted from ?/ in 

 Table I. 



Table XIV. is formed by subtracting the values of ^h and Sh during the revo- 

 lution of Table XV. from the values of the same elements 164.78 years earlier 

 or later. Or, we have 



AP,.,z= — 2{8h' — 87io) 



8h' and SJif representing the values of Sh and Sk at any epoch, and 8ho and Sk^ their 

 values at that date of the period 1779-1943 when the planet had the same mean 

 longitude as at the epoch in question. 



The sum of the sixteen quantities P.^ sin I, P^ cos 1, P^.o sin 2 I, P^^. cos 2 I, hv (i to gj , 

 I, y, and the equation of Table XV. will give the true distance of the planet 

 from its ascending node, which we represent by u. 



Table XVI. gives the reduction to the ecliptic for the years 1800, 1900, and 

 2000, together with the change of the reduction for a century. The constant 



60" 



has been added to render all the numlicrs of the table positive. 



