Mathematical investigations concerning the laws of the equilibrium of fluids analagous to the electric fluid / by George Green.
- George Green
- Date:
- 1833
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.
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![spheres only are to be considered, the resulting formulge, as we shall afterwards sliow, will be much more simple if we expand the density p in a series of functions similar to those used by Laplace {Mec. Cel. Liv. iii.): it will however be advantageous previously to demonstrate a general property of functions of this kind, wdiich will not only serve to simplify the determination of V, but also admit of various other applications of xla. Suppose, therefore, is a function of 6 and sr, of the form con- sidered by Laplace {Mec. Cel. Liv. iii.), r, 0, 'sr being the polar co-ordi- nates referred to the axes X, Y, Z, fixed in space, so that X — r cos 0, y = r sin 6 cos -sr, x = r sin 0 sin •ar; tlien, if we conceive three other fixed axes X^, Y^, Z,, having the same origin but different directions, Y^''> will become a function of 9^ and 'ar,, and may therefore be expanded in a series of the form (6) = FW + F/') + F® + F® + &c. Suppose now we take any other point and mark its various co-ordinates with an accent, in order to distinguish them from those of jy; then, if we designate the distance pp by {p,p), we shall have ,—1 — 2rr' [cos 9 cos 9' + sin 9 sin 9' cos (w- — •ar')] + r'H {p,p) i = i (O'’ + Q’ • ^ + 0“’ 7 + O'” , as has been shewn by Laplace in the third book of the Mec. Cel., where the nature of the different functions here employed is completely ex- ])lained. In like manner, if the same quantity is expressed in the polar co- ordinates belonging to the neAV system of axes X^, F,, Z,, we have, since the quantities ?' and r are evidently the same for both systems, = - (Q,™ + Q,<'> - + Q,'”-. + + &C. ; {p,p) r \ r 7 r’ ) and it is also evident from the form of the radical quantity of which](https://iiif.wellcomecollection.org/image/b28754207_0018.jp2/full/800%2C/0/default.jpg)
