518 
Proceedings of the Royal Society of Edinburgh. [Sess. 
The small eccentricity found by Dr Struve is neglected. From these 
elements the auxiliary quantities B, P, and U are found by means of 
the equations : 
cos B sin P = — cos (a - N) sin I 
cos B cos P — sin 8 sin (a - 1ST) sin I 4- cos 8 cos I 
sin B = cos 8 sin (a - N) sin I - sin 8 cos I 
cos B sin U = — cos 8 sin (a - 1ST) cos I - sin 8 sin I 
cos B cos U = cos 8 cos (a - N), 
where 
P = position angle of the pole of the orbit of the satellite 
B = planetocentric latitude of the Earth with reference to the orbit of the 
satellite 
180— U = planetocentric longitude of the Earth with reference to the orbit of the 
satellite. 
With these auxiliary quantities the distance 8 and the position angle p are 
found from the equations : 
8 sin (p - P) = r sin (u 4- U) 
s cos (p — P )=r sin B cos (u + U), 
where 
r — a(p)/p 
P =(/>)(! + a cos B cos (u + U) sin 1") 
and (p) is the mean distance of Neptune from the Earth. 
The formulae employed in obtaining the corrections to the elements are 
those deduced by Mr Marth, viz. : 
s sin dp = r sin r sin du 4- (?’ sin r cos I + r cos r cos u sin I) sin dls 
— r cos t sin u sin dl - r sin r cos u . 2e sin Q 
+ r sin r sin u . 2e cos Q 
ds = r cos cr cos r sin du 4- r cos cr sin p cos 8 sin cZN 
+ r cos or sin t sin u sin dl 
2e sin Q 
— r cos cr cos t cos u + - sin u 
2 
+ (r cos a cos r sin u — -- cos . 2e cos Q + , 
V 2 ) a 
where 
e = eccentricity 
Q = longitude of periastron measured from the node of the 
satellite’s orbit on the Earth’s equator 
sin r = - sin B 
s 
V • ~T‘ 
cos r = - cos B sin {u + TJ) 
S 
COS O' = cos B cos (u 4- U). 
