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.