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.

62 PLANETARY THEORY.
rj d dR
min x, -( 2 - =i X,
(x2 + y2) dmlx
Ayy dRl
(x2 + y2)" dy
79. PROP. 30. To transform the equations for rectangular co-ordinates into equations for polar co-ordinates.
The differential equations of motion are,
d2, d2y
dt2 dt2
dx dy
Multiplying the former by 2 -, the latter by 2-9 and
dt dt
adding them, and putting for X and Y their values,
d ld2 dy2 2t dx dy\
0 dt=, i 2 d t( + (,2 + y2) \d' + Y dt
fdR dx dR dy\
+ 2, -. Tt + _._
\d~ * dt dy dt I
Again, subtracting the product of the first by y from that
of the second by x,
d ( dy dcx\ dR dR
o= ~ —~ - y1 — ~ +. x-t - ya dt tY dtt dy
Let r, be the true radius vector of m: 0, its true
longitude; r' and O' those of '.
TheV n x = r cos O,: y = r sin 0,: = r'cos ': y'= r'sin 0':
and these equations become,
d | dr\;2 2,d 2 2 dr (dR dx dR dy
dt dt r +ri +2 1.-+ I
d o t u dt r dt dx dt dy d ti
dtl 'dt: dyd.d
80. Now it must be remarked that the only way in
which R can depend on the position of m is by depending
on the two co-ordinates which determine thé place of:m.