THE ORBIT OF NEPTUNE. 41 



The other terms in the longitude, logarithm of r, and latitude, representing the 

 mean longitude of the planet by the initial letter of its name, are : 



: — 2".949s: 



— 9 .942 s 



— 1 .907 s: 



— .610 s 



— .237 s: 



— .104 s: 



— .041 si 



— .017 s: 



— .007 s: 



n A — 0".002cos A 6r^= 314 cos A 



a2A — 0.094cos2^ 4-162cos24 



nSA +0.016 cos 3^ + 38 cos 3^ 



n4.4 -fO .004 cos 4 4 + 13 cos 4^ 



a5A + 5 cos 8^ 



n 6 ^ + 2 cos 6 ^ 



nlA 

 il8A 

 adA 



+ 18". 552 sin [S — N') + 397 cos (S—N) 



— .137 sin 2(5 — i\^) " + 4cos2(5 — iV) 



— .012 sin 3 (5 — i\^) 



— 0".524cos(2S— iV) + 10 sin (2 5— iV^) + lco3(25 — iV) 



— 0.058sinS' +0.047 0035 + 4sin(S — 2i\r) + 4 cos (5 — 2 i\^) 

 + 0.166sin (S—2N-) — .436 cos (S— 2iV) + 701 cos (7"— iV) 

 + 34 .121 sin (J— N) + 4sin(2/— iV) + 18 cos (2 J" — ii^) 



— .011 sin 2 (J"— N) — 5 sin {J— 2 M) + 4 cos (J"— 2 iV^) 

 + 0.783sin(2/— N) — .164 cos (2/— iV) 



— 0.101 sin 7" +0.097 cos 7" 



+ .826sin(J" — 2iV) + .297 cos (J"— 2iVr) 



(5|3„ = — 0".302 sin 5 + O".005 cos S + 0".041 sin J + 0".563 cos J. 



It will be observed that in the perturbations of the longitude by Jupiter and 

 Saturn we have neglected a number of small terms, the coefficients of the four 

 largest of which are each about 0".03. The probable error in the theory pro- 

 duced by this neglect is 0".04, and it was judged best, therefore, not to encumber 

 it with them. But, should any one wish to include their effect, it can readily be 

 calculated. Then, we have 



Provisional longitude of Neptune, referred to the mean equinox 



z=: Precession, + Longitude in pure elliptic orbit, from elements page 39 



+ ^/ -1_ (/>^ + 2 87c} sin I + (P,.i — 2 Sh) cos I + P,,^ sin 2 Z + P,^ cos 2 Z + &„ 



+ Eeduction to ecliptic. 



Common logarithm of the radius vector 



=z Log. radius vector in elliptic orbit 



— .0005920 -\-8a-\- {R,,^ — Mhh) sin I + {R,,^ — Mhh) cos I + h;. 



Latitude r± 



Latitude in elliptic orbit (the longitude being increased by the perturbations), 



+ {Bs.x + hq) sin v -\- {B^^ — §2^) cos v + 5/3„. 



I is the mean longitude of Neptune, and v its true longitude in orbit, referred 

 to the mean equinox of 1850.0. 



§ 20. These forinula3 give the following heliocentric positions of Neptune : 



6 May, 18G5. 



