Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

ORBrIS SUPPOSED INCLINED. 121
Then
u u
e = %/(2 + v2)~ tan r = - e' =,/(u'2 + v'2): tan'= -,.
U U
A similar process is to be used when the secular variations
produced by the mutual perturbations of several planets are
to be calculated.
On examining the geometrical meaning* of these expressions it is easily seen that the motion of perihelion is'
on the whole progressive; and that its *mean period is
ai
2Tr
if p, be greater than P2, and if P2 be greater than pl.
~2
Similarly the motion of perihelion of m' is on the whole
progressive; and its mean period is - if L be greater
than P - or if 2 be greater than P1
X2 a2 XT2
145. We shall conclude with shortly describing the
process to be followed when the orbits of the planets are
inclined. If we put
m' m (x'x l+ y' y + 'z)
{ (' _ - x)2 + (y -y)2+(z.z)~ + (x' y + y' +z' )t
equations similar to those of (77) may be found. If we
take for xy the plane of the disturbed planet's orbit at a
given time, z is only the effect of the disturbing force and
may therefore be neglected in R. And all the equations
for the perturbations of radius vector and longitude, or for
the variation of the major axis, &c. hold equally well: the
only difference in the process is, that for R we must
expand
m m' ( 'T + y y)
* The center of each orbit describes an Epitrochoid round the focus which
coincides with the Sun's center. This will be seen on comparing the expressions
for u and v (which are rectangular co-ordinates, referred to the focus) with the
expressions for x and y in Peacock's Examples, p. 193. The circles must be
supposed to revolve uniformly.