102 ON THE SELF-INDUCTION OF [364
purpose this requires transformation, so as to express the inductance in terms of the situation of the elementary circuits relatively to the circular axis. In the figure, 0 is the centre of the circular axis, A the centre of a section B through the axis of symmetry, and the position of any point P of the section is given by polar coordinates relatively to A, viz., by PA (p) and by the angle PAG (fa. If p1} fa; p2, fa be the coordinates of two points of the section Plt P2, the mutual induction between the two circular circuits represented by P1} P2 is approximately
I2 - ) i Pi cos fa + pz cos fa , p-i + PZ + 2pi3 sin2 fa + 2p2a sin2 fa ~a~{ 2a 16a2
i>n) + 4tplpz sin fa sin fa] , Sa flog —
pi COS (^i + p2 COS ^>a
2o :
, x2 + p2-} - 4 (pj2 sin2 <k 4- p22 sin2 02) + 2pxp2 cos (^ - </>2) +--------------------------------._ , (2)
in which r, the distance between Pa and P2> is given by
r- = px2 + p22 - 2PJP2 cos (0! - ^9)......................(3)
Further details will be found in Wien's memoir; I do not repeat them because I am in complete agreement so far.
For the problem of a current uniformly distributed we are to integrate (2) twice over the area of the section. Taking first the integrations with respect to (j>l} <f>2, let us express
of which we can also make another application. The integration of the terms which do not involve log r is elementary. For those which do involve logr we may conveniently replace <£2 by fa + 0, where 0 = <j£>2 — fa, and take first tjie integration with respect to <£, fa being constant. Subsequently we integrate with respect to fa.
It is evident that the terms in (2) which involve the first power of p vanish in the integration. For a change of fa, fa into TT — fa, Tr — faiven by Maxwell f and Sellmeier, the resonating bodies take their motion from those parts of the sether with which they are directly connected, but they do not influence one another. In such a case the boundary conditions involve merely the continuity of the displacement and its first derivative, and no complication ensues. When there is no refraction, there is also no reflexion. By introducing a mutual reaction between the resonators, and probably in other ways, it would be possible to modify the situation in such a manner that the boundary conditions would involve higher derivatives, as in the case of the stiff string, and thus to allow reflexion in spite of equality of wave- velocities for a given ray.