ON THE MOTION  OF PENDULUMS.                           39
By virtue of (67), udy - vd% is an exact differential.    Let tlien udy — vdx = dx ......................... (68).
Eliminating p by differentiation from the two equations (66), and expressing u and v in terms of % in the resulting equation, we get
d*d*      ld\d*
"^0 ............ ^(69)'
and, as before
% = %i + %_2 .......................... (70),
where
<^71>-
We get from (66) and (68)
d I f     d*     1 d
d (ff      d*      1  d •dx^^-^
which becomes by means of (70), (71), and (72)
^
dx  ^y    dt dy
26. Passing to polar co-ordinates r, 6, where d is supposed to be measured from the axis of x, we get from (68), (71), (72), and (73)
JBrdd-®dr = dx ........................ (74),
Id     1 ^+ f^ ^+ ?
Id    Id"     1 d
dt \ dr
..................
^    ^J
R, @ in (74) being the velocities along and perpendicular to the radius vector.