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.

PERTURBATION OP RADIUS VECTOR. 67
The equation now becomes
d_(r + 2 rSr) 1 ir 4 d(R) dR
d2(r~2r -=)2C+2p — 4 ' --- —r.. —.
dt2 \r 2C+2-) J dt dr
If there had been na disturbance, the equatian wauld
have been
d2. r2 2,i
d- =2C +dt- r
where r has the same value as in the last equation.
Taking the remaining parts which relate to the disturbing
force and the disturbance,
d2(rSr) îr d(R) dR
d(r- --— =- - — 4f
dt2 r2, dt dr
d2 (rir), d(R) dR
dt2 r3 r dt dr =
87. PROP. 32. To solve the equation for the perturbation of the radius vector,
Let* a be the semi-major axis of the approximate orbit
of mn: e its eccentricity: - the longitude of the perihelion:
e the mean longitude of m when t=: n= ( U).
* The equation may be completely integrated in the following manner. In
an undisturbed orbit,
2X =, d2.
dt2 r3 dt2 r3
The form of these equations is nearly similar to that for ror. Assume therefore
rSr =Mx + Ny: and as there are two quantities M and iN to be determined,
assume
d (r r)M d+ d
dt dt dt
andet2fd(R) dR
and let 2 ( ---d-t- + r d= S.
t d(r r)
Then from our last assumption, (since the complete expression for d(r is
dx dM dy dN\
Mdt +xct +Ndt +Yt t)
dM dNV
+dt y =0dt: