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Full text of "Scientific Papers - Vi"

34          ON  THE  MOTION  OF  SOLID  BODIES THROUGH VISCOUS  LIQUID      [354
so that
^*                             ^t
dx J o
\/t
o (t  ry                                              o            o
- r2) rdr = TT {< () - 0 (0)},
o
where r2 = x2 + 2/2.
Now, if t' be any time between. 0 and t, we have, as in (19),
Multiplying this by (t  t')~^ dt' and integrating between 0 and t, we get
I ' V'MM + ?.^ f '    dt>     !' V'^dr '   (^-0*       ^ ^ (t-rflo   (t'-r)^
=                .....       (22)
'
In (22) the first integral is the same as the integral in (19). By Abel's theorem the double integral in (22) is equal to TrV(t), since F(0)=0. Thus
(23)
If we  now eliminate the integral between (19)  and  (23), we  obtain simply
dV    4>p2v Tr           kpv-     .                                 ._,..
____          '         I/  r, _     i        n , I4-                                           I O/L\
~77                  '     a         -----1" y  \ "    '....................V        /
as the differential equation governing the motion of the lamina.
This is a linear equation of the first order.    Since V vanishes with t, the integral may be written
in which t' = t. 4p2j//<r2.    When i, or t', is great,
sotbat                           =         _1+             !         +...   ............. (27)
g'o-2        VTT           v                                              ^    '
Ultimately, when t is very great,n ingenious application of Abel's theorem Boggio has succeeded in integrating equations which include (19)*. The theorem is as follows:  If ty(t) be defined by