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.

PERMANENT VARIATIONS OF ELEMENTS. 117
m'C(<) m' ae d C,( m'a' e' d C(
2 4 da 4 da'
m a2e2 dC(<) m''"e'2e dC
8 da2 8 da'2
(where the two last terms are introduced by sin2 nt + e —r&c.)
- m' C(). eecos (' - ar')
ee' ( dCix d C(l)
-m'. - a — - + a ) cos( - )
2 \ da a da
ee' d' C(<l
- m aa. co — s (zer - ~r)
4 dadad
(by expanding cos 0 - O')
and other smaller terms.
By the use of the formula of (127) and (128) these
collected terms rnay be put under the form
mn'C(), D') aa' D(2 aa'
- -- - -m. --- (e2'+ e'2) + m' -- ee cos (- - ').
2 8 4 '
141. PRoP. 57. To examine the terms which produce
permanent variations in any of the elements.
The terms of R which depend on t are always in the
form of a cosine with a constant coefficient. These terms
then \as will be seen on referring to (118)1 can produce
only periodical variations in the elements. It might at first
sight appear that there is in e, a term multiplied by t, but
{as in (119)} it will be seen that there is always ini nt a
similar term which destroys it. It will be sufficient therefore to examine thé constant terms found above.
dR
(1) With regard to these terms, d = (since e is not
~b~ dglâ
da,
found in them): therefore there is no constant term in
dt
or no permanent variation in the axis major. Similarly
there is no permanent variation in n.