Mathematical investigations concerning the laws of the equilibrium of fluids analagous to the electric fluid / by George Green.

  • Green, George, 1793-1841
Date:
1833
Catalogue details

Licence: Public Domain Mark

Credit: Mathematical investigations concerning the laws of the equilibrium of fluids analagous to the electric fluid / by George Green. Source: Wellcome Collection.

    12/64 (page 12)
    retains the same numerical value, and merely changes its sign when we pass from the element dv to a point diametrically opposite, where the co-ordinates x, y', z are replaced by - x', - y', —z': it is easy to see that the function F”,, depending upon pi, possesses a similar property, and merely changes its sign when x, y, z, the co-ordinates of p, are changed into - x, -y, —z. Hence the nature of the function F is such that it can contain none but the odd powers of r, wdien we sub- stitute for the rectangular co-ordinates x, y, z, their values in the polar co-ordinates r, 9, tzr. Having premised these remarks, let us now suppose F is divided into two parts, one F/ due to the sphere B which passes through the particle p, and the other F/' due to the exterior shell S. Then it is evident by proceeding, as in the case -where p = (1 -that F,' will be of the form F/ = \A -H Br^ + Cr‘ + &c.|; the coefficients A, B, C, &cc. being quantities independent of the variable r. In like manner we have also F = fr^dr’de'dsr' sin 0' (1 - . /v// (F') g^-”; the integrals being taken from r=r to r = l, from 9' = 0 to 0' = 7r, and from cr' = 0 to -za-' = 2 tt. By substituting now the second expansion of g^- before used (Art. 1.), the last expression will become F, = 7; + Ti+ 7; + 7; + &c. of which series the general term is T, = fd9'd7!r' sin 9' Q, fr^-‘dr'(I - ry ^ ^|. (ry Moreover, the general term of the function \// (r'*) being represented by Atr-‘, the portion of 7] due to this term, will be (a) 7'^ fd9'd-sr’ sin 9' (1 — 7-y the limits of the integrals being the same as before.