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.

CALCULATION FROM DISTURBED ELEMENTS. 99
120. PROP. 47. To calculate the radius vector of a
disturbed planet.
This must be done by substituting for a, e, &c. in
the expression
e~
r, = a, { + - + &c. + (e,+ &c.) cos (nt +, - a,) - &c.}.
Proceeding in the same manner as in the last Article, and
using the same instance, we find for the additional terms in r
26n a2
26na- 8 1-Pe'e'2 cos 13 (nt + e) - 8 (n't + e') - 3 w - 2ar';j(iSn - 8n')
3na2.i3n -- 8tj') P ee'2 cos 12(nt + e) - 8(n't+ ) - 2r-2 'r'.
The latter of these is generally the nore important.
DEVELOPMENT OF R.
121. PROP. 48. Supposing (a- 2 aa' os o + a2)
< (a2 - 2 aa 'cos w + a")
expanded in a series of the forin
2 C) + C + cos w + Ccos + &c.:
to calculate numerically the values of C() and C(').
Case 1. Let a and a' be very unequal: suppose for
instance that a' is Inuch less than a. Then the fraction
1 ( a' a''a -
=-1-2-cosw+O +
a a a
For 2cosw put e'V —, + e-W/- (e being here the base of
Napierian logarithms) then the fraction
a a a
1 la' 1-.Sa2 1.. a'3
e ~x/-1 a4 -1- 1.3a~ (/ —e- + s/
=-si _ V -+ - î - - + IV/ -e + e.6 +&c.}
a t 2a 2.4a 2. 4. 6a3
7-2