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.

IMPORTANT TERMS IN PERTURBATION OF RADIUS VECTOR. 73
there will be one term P. cos (nt + Q), introduced by the
first term of v. The equation for rSr, considering this
term only, will be
0= d + n2. rrr+P.cos(nt+Q).
d t2
The solution of the equation in this form, as in (5) and (43),
would have terms depending on an arc, which would admit of
unlimited increase, and therefore our original supposition of
elliptic motion would not be even an approximation. T'o avoid
this difficulty, we must, as in (44) and (44'), suppose the first
term of v to depend on cos c(nt+e- ar), which will change
the term in the equation to P cos (cnt + Q). This amounts
to supposing, as in (61), that the perihelion has a constant
motion. We shall not in this place proceed further witli
the investigation of its motion, as it will be better found by
the method of variation of elements, to be explained hereafter.
There will also be one constant term in r'r: this shews
that, the mean motion being given, the axis major is not the
same as if there were no disturbing force.
92. PROP. 33. To examine the terms of R which are
increased most in integrating the equation for r2r.
It is evident that the terms which are most increased
by the integration are those in which
(pn - qn')2- n, or (p + 1)n - qn'.(p - 1)n - qn',
is small; that is, those in which one of the quantities
(p + 1)n - qn', (p -1)n - qn',
is small: where p and q are whole numbers, and may be positive or negative. Consequently, if we can find two numbers
p + 1 and q, or p - 1 and q, which make
(p + l)n - qn', or (p - l)n - qn'
very small, we may expect that the term of 3r depending
on cos (pnt - qn't + Q) will be large. The train of teims
introduced with it will all bc affected by the same divisor,
and therefore, though multiplied by e, e2, &c. may be sensible.