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.

110 PLANETARY THEORY.
dR dR
132. PRor. 53. In the expansion of R r, r -dr, d
dl dr da
dR dR
-, and tle principal coefficient of a term
de d -
cos (pn - qn')t + Q
will contain powers of e and e' to p - q dimensions*.
(1) If we examine the expressions for v and v' it is
immediately seen that a term cosg(nt + e - ar) has for coefficient a multiple of eg, and that a term cosh(n't+e' - ')
has for coefficient a multiple of e'h, &c.
(2) In any power of v, each term is either a power h of
eg.cosg(nt+ e- r) or a product of e9.cosg(nt+ e - )
by eh. cos h (nt + e - -). The first of these will produce
eg. co coh (nt i+ -) +B cosg.(h-2)(nt + E-~-) + &c.}
where eg^. B is not the principal term in the coefficient of
cosg. (h - 2)(nt + e - ar), since lower powers of e will be
found multiplying that cosine, in the expansion of the power
h - 2 of eg. cosg(nt + e - '): but where egh is the lowest
power in the coefficient of cosgh(nt + c - r) since it las
been produced by the simplest combination which could
form that cosine. The second will produce
e9+khcos (g ) ( + h) (t - ) cos (g- h) (nt + e - )}.
The first term is produced by the simplest combination of
arcs which could form it; the second is not, for it might
be produced by the product of
eg-21. cos (g - 2h) (nt + e - wr) and eh cos h (nt + e - r),
which would have for coefficient eg-9. In all these terms
then we have this general rule: the principal coefficient of
cosxo(nt + e - ar) is a multiple of eJ.
(3) A similar proposition applies to the terms of the
expansion of any power of v'.
* This is also true if the orbits be inclined: the inclination being supposed
of the same order as the eccentricities.