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.

VARIATION OP PARAMETERS. 83
solution of the unaugmented equation. Assuming then
'= A'. cos (nt - B'), we have
dz' dA'
- = n A'. sin (nt - B') + cos (nt - B'). d
dt dt
dB'
+ A'. sin (n t - B') dt
dz'
But = -nA.sin(nt - B): and is to have the
dt dt
same form: therefore we must have
d z'
dt
— =- naA'. sin (n t - B').
Consequently the remaining terms of the complete expression
dz'
for -d must =0; or
dt
dA' dB'
0 = cos (nt - B').- + A' sin (nt - B'). dt
dz'
Differentiating -dt and substituting in the given equation,
dA' dB' a
0= - sin (nt- B). + A'. cos(nt -B'). -+.cos(pt-Q).
dt (t +n
From this equation and the last we find
dA' a
d' —. sin (nt - B'). cos (pt - Q),
dB' a
d = - n A' cos (nt - B). cos (p t - Q).
The values of A' and B' will be readily found, if we suppose their difference from some constants A and B to be
so small that in the terms multiplied by a we may put
A for A', and B for B'. Then we have
z= A' cos (nt- B'),
where A'= A - a cos (nt + pt- B-Q)
2n(n + p)
a
2 an(n cos(nt -pt -B+ Q).
2n(n7 -p)6
6-2