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Since (5) is a differential-equation of the second order, its sol connected in a well-known manner.    Thus
0#S9    «<fcSfi_0
8ld?~   **?~   "......................
and on integration
5i^_&^' = constant = i,  .................
1 dy         dy
as appears from the value assumed when T? = 0.    Thus
which defines £> in terms of $lt
A similar relation holds for any two particular solutions.    For t
The difficulty of the stability problem lies in the treatment of the condition
in which %, 7/l5 and X are arbitrary, except that we may suppose rt be positive, and ^ negative. In (31) we may replace e^, e~A7?b sinh X?? respectively, and the substitution is especially useful when of integration are such that 7/1 = — ^2. . For in this case
[i>* S cosh XT? dy = 2       s cosh XTJ dy,
J o fls                                          pa
$ sinh XT? O!T? = 2i I    < sinh XT? cfo;;
J n,                                             J I)
and the equation reduces to
la                                       /-Us
S] cosh \ydy.\    tn smh XT? C^T? .' o                         ."' o
r%                          PJ
-      s2 cosh XT? rtT?.      ^ sinh XT? cfo? = 0,
Jo            _               Jo
thus assuming a real form, derived, however, from the imaginary tei In general with separation of real and imaginary parts we have by the real part
f                        f                  f
ST. e^ dr).lsz e~Xy> dy -  ti e^ dy . It, e~
s2 e^ dy +' ^ e~^ c^ . k eAl» C^T? = 0, .
* Kather to my surprise I find this condition already laid down in private papers dtions, of (5) are worthy of notice. If S= s + it, we have