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.

20 LUNAR THEORY.
dx d2'. dy d'y 2P dx dy\
Hence 2 -. — + 2. — -d + y )
dt dt2 dt dt? p dt dt
yT I dy dx\
+ -- - y -.
p dt dt
dxo d2 x dy y d2y d d2 [dy 2
But 2.o. —2..+ I
dt dt2 d+ t dt' dt t dt dt)
d fdpY dO\1. dx dp. d
+P-2 5 since = — cos 0 - p sin 0 -dt \dt) ' \dt dt dt dt'
dy dp dO dx dy dp
- = d sin 0 + p cos And x " dp = d;
Ft dt dt 'y dt
dy dx dO
Also xdt -y 2 T -p Tp.
d2y d2 d d dy d d d\
But x - - - y P. -
t d tt dt Y dt) dt dt);
d (p. d) = Tp..........(b).
And (p) - -S....... (c)
d t2.....
31. These equations (a), (b), (c), appear to be the
simplest equations to which the motion of a point can be
reduced. In their present form, however, it is not possible
to integrate then: we must obtain equations between p, 0,
and s, independent of t. This is the object of the next
proposition.
32. PRoI. 11. To eliminate t from the differential equations.