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.