ON  THE  MOTION  OF  PENDULUMS.
131
second, up to the order i, as well as ^r (#, t) itself, vanish when # = 0. Substituting now in the second equation (8) the expression for v given by (184), we see that the equation is satisfied provided
3 +«•«*-«.   f
These equations give
<£ <X, t) = x 00 e"^     t « 0 = 0- 0*0
where ^, a- denote two new arbitrary functions. Substituting in (184), and then passing to the first of the equations of condition, we get
0 = x(aO + er(aO, whence cr (x) = — % (%) and
o  r«5 /*oo 0 = - I         cos a (# + x) e ~ ^ % (xf) dx da
71" J 0   J 0
I        /•
= -7=-
VTT/JitJ
i?
(185).
The second of the equations of condition requires that
V= ~jL= r e""fe x 00 *»' = -T- f  e~*2 % (25 V/z7^) ds. W/tfJo                          V^A        ^V       ^
Since the second member of this equation must be independent of t, we get x 00 = a constant, and this constant must be equal to F, since
2 r .# ,
-j- I   e     * Substituting in (185) we get
7 r
fl =-------.        €
VTTLi't J 0
.(186).
For the object of the present investigation nothing is required but the value of ^~ for x = 0, which we may denote by ( -y- ] .    Wo
G>&'                                                                                 \vix/ q
get from (186)
dxj
.(187).
Now suppose the plane to be moved in any manner, so that its velocity at the end of the time t is equal to f(t). We may evidently obtain the result for this case by writing/' (t')dt' for F,
9—2