a table of the values of the density at different points on the surface of the plate, calculated by means of the fornula (29), together with the corresponding values found from experiment. We thus see that the differences between the calculated and observed densities are trifling; and moreover, that the observed are all something smaller than the calculated ones, which it is evident ought to be the case, since the latter have been determined by considering the thickness of the plate as infinitely small, and consequently they will be somewhat greater than when this thickness is a finite quantity, as it necessarily was in Conlomb's experiments. It has already been remarked that the method given in the second article is applicable to any ellipsoid whatever, whose axes are a, b, c. In fact, if we suppose that I, y, are the co-ordinates of a point y within it, and I', y, z' those of any element do of its volume, and afterwards make q=a.cose, y=b.sin & cos , :=c. sin & sin , t'=a.cos, y'=b.sin 8' cos s', é'=c. sin 6 sin s', we shall readily obtain by substitution, V = abc sp.rdr'de de'sin 6. (Add-Qurr' + 27 ; the limits of the integrals being the same as before (Art. 2), and Are cos + * sin cos s' tosin sin s', =ofcos 8 cose + 'sino sin coss cos s'tasin o sin 6-sins sins', medicos 8* +* sin 6* cos s + e* sin o sin ". Under the present form it is clear the determination of V can offer no difficulties after what has been shown (Art. 2). I shall not therefore insist upon it here more particularly, as it is my intention in a future paper to give & general and purely analytical method of finding the value of V, whether p is situated within the ellipsoid or not. I shall therefore only observe, that for the particular value the series Ui+ 0 + 0 + &c. (Art. 2) will reduce itself to the single term U., and we shall ultimately get which is evidently a constant quantity. Hence it follows that the expression (30) gives the value of p when the fiuid is in equilibrium within the ellipsoid, and free from all extraneous action. Moreover, this value is subject, when n < 2, to modifications similar to those of the analogous value for the sphere (Art. 7). . ON THE DETERMINATION OF THE EXTERIOR AND INTERIOR ATTRACTIONS OF ELLIPSOIDS VARIABLE DENSITIES. * From the Transactions of the Cambridge Philosophical Society, 1835. (Read May 6, 1833.) |