438 NEW OBSERVATIONS OF THE PLANET MERCURY. 
and equate it to that of the circle supposed described about that focus with the 
+ 
length of radius "^ ah. This geometrically is the point of intersection of the two 
of r is common to both 
Consequently for the point sought 
'sjah 
Since a may be taken as unity, this is 
^/h 
or cos V 
or since 
V" 
(1 - o 
" 1 + fi COS V 
(1 
-e^) 
1 + 
e cos v^ 
1 
• 
,1 » 
1 
e^ (1 i 
2\l r ^ ,.i\\ 
h /-. 1 1 _ e2 _ (1 _ e'^Y ( 1 - eY - 1 
= yi — e^, cos w = i L =. i '_ 
In the case of Mercury, e = .205605 ; v, the true anomaly of the point of maximum 
libration is therefore 98° 55'.13. 
But -—-= cos u, where u is the eccentric anomaly ; and u — e sin u = ^, where 
t, which is the amount of the maximum lib 
^ is the mean anomaly 
lion, is 23° 40' 38". 
As the period of the planet is 87.96926 days, the mean angular velocity of revolu- 
F 
tion, which in the case of Mercury is also the angular velocity of rotation, is 4°. 092 a 
day; the angular orbital velocity at perihelion, 6°. 346 a day; and the angular orbital 
velocity at aphelion, 2°.755 a day. 
Consequently the daily gain of the angular orbital over the angular axial velocity 
or the daily increase of libration east or of decrease west— at perihelion is 2° .254; 
similarly the loss at aphehon is 1°.337. 
The time taken by the planet from perihelion, where the libration is nothing, to 
the point of maximum libration, is 18 days, 9.28 hours, and from this point to 
aphelion 25 days, 14.35 
As the planet's heliocentric longitude at perihelion is now 75° 51', the heliocentric 
tude of the points of maximum libration are 174° 46' and 330° 56', while at 
5° 51' and 255° 51' the libi 
g 
Calculating the libration, together with the phase angle for February and 
March, 1897, — as this will be the most useful to us later on in the paper, 
we get 
