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Full text of "Scientific Papers - Vi"

158                             ON  THE  APPROXIMATE   SOLUTION   OF                            [374
or by use of (3)
, _ +y +       -     _      + .
_
""        36        da?       dx-\y}           120       d
The evanescence of ^ when ^ = 0 may arise from this axis being itself a boundary, or from the second boundary being a symmetrical curve situated upon the other side of the axis. In the former paper expressions for the "resistance" and "conductivity" were developed.
We will now suppose that -v|r = 0 along a circle of radius a, in substitution for the axis of x. Taking polar coordinates a + r and 6, we have as the general equation
-                    -         2        .
0 ................... (5)
Assuming                       ty = J^r + It*r- + jft3?'3 + ... ,........................(6)
where Rl} R2, &c., are functions of 6, we find on substitution in (5)
2alR2 + aRl = 0,
^.ftr-+....................(8)
is the form corresponding to (2) above.
If ^ = 1, (8) yields
1      1        r-        r-   d- fl\
(9)
r
expressing ^ as a function of 6, when r is known as such.    To interpolate a curve for which p takes the place of r, we have to eliminate 1^ between
,                              ,      ,,           p
and                            ^ = E3p - --
r      2a                a
Thus                P = T*~^ (Pr2 ~ ^ + ^
and by successive approximation with use of (9)
1.2'  0, and then no impressed bodily forces are called for either at rest or in motion.