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.

64 PLANETARY THEORY.
82. For changing the other equation we must remark
that, as R is now to be considered a function of r, and 0,
both of which depend on x and y, we must put
f dR dR dr, dR dO
for d -* + -0
d' dr 'dx dO' d~;
dR dR dr dR dO,
and for —, - + ' — +;
dy dr, dy de, dy
dr dO,
where in obtaining d, ' &c. we must be careful to
d-ro d~x'
express r, in terms of x and y only without 0,: and 0, in
terms of x and y only, without r,.
dr
Thus r' =2+y2. 2r =2;
'dx
dr, x dr
o r ' - = cos. Similarly ' - sin 0
d. r ' dy
And tanO=-Y; whence 1 dO- - y in0,
x Cos2, dx x2 r, cos 2
dO, sin 0,
or -- -
dx r
I dO, 1 1
Similarly. = - =
cos e0 dy x r cos 0
d e0 cos 0
whence -d, = -cs
dy r,
dR dR dR sin0,
Consequently d = cos 0 - d. --
d-X dr, c d6 r
dR dR.0 dR cosO
d- dR sin0, + ---
dy dr, d0, r
and the second equation becomes,
d ( 2 dO dl
o' ' dt) d
dt ''dt d0l'