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 : 



Sv, = 2". 949 sin A 0".002 cos A 6r,= 314 cos A 



9.942sin2J. 0.094cos24 + 162cos2X 



1.967 sin 3 A + .016 cos 3 A + 38 cos 3 A 



0.610sin4.4 +0.004cos44 + 13cos4^1 



0.237 sin 5 A -f 6 cos 5 A 



0.104sin64 + 2cos64 



.041 sin 7 A 



.017 sin 8 A 



.007 sin 9 A 



+ 18".552sin (S N) + 397 cos (S W) 



- .137 sin 2 (S;/) + 4 cos2 (S JV) 



.012 sin 3 (S #) 



0".524cos(2S N) + 10 sin (2 S N) + lcos(2S JV) 



- 0.058sin5 + 0.047cosS + 4sin(S 2N) + 4 cos (S 2 JV) 

 + O.lGGsin (S 2N) .436 cos (S 2 N) + 70lcos (JN) 

 + 34.121sin (J N) + 4sm(2JN') + 18cos(2^ N) 



- 0.011 sin 2 (J N) 5nin(J2N) + 4 cos (J 2N) 

 + .783 sin (2 J N) .104 cos (2J JV) 



.101 sin J" + 0.007 cos J 



+ .326sin(^ 2N) + .297 cos (J 2 If) 



<5/3. = 0".302 sin S + 0".OG5 cos S + 0".041 sin J + 0".5G3 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 

 rr Precession, + Longitude in pure elliptic orbit, from elements page 39 

 + M + CP..I + 2 3&) sin Z + (P tl 2 3A) cos I + P,. 2 sin 2 1 + P c , 2 cos 2 1 + Sv, 



-f- Reduction to ecliptic. 



Common logarithm of the radius vector 



= Log. radius vector in elliptic orbit 

 - .0005920 + 8a + (fl tl MSK) sin I + (R^ MM) cos I + 8r . 



Latitude 



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



sin v + ,! 8 cos v 



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

 to the mean equinox of 1850.0. 



20. These formulae give the following heliocentric positions of Neptune : 



6 May, 1865. 



