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and boy being thus given, Bl") and consequently the second line of the expansion (25) are also given.
From what has preceded, it is clear that when V is given equal to any rational and entire function whatever of x and y, the value of f (ac', y) entering into the expression
p= (1 – 3) 7.5 (a", y'), will immediately be determined by means of the most simple formulæ.
The preceding results being quite independent of the degreo 8 of the function f (x', y') will be equally applicable when : is infinite, or wherever this function can be expanded in a series of the entire powers of só, y, and the various products of these powers.
We will now endeavour to determine the manner in which one fluid will distribute itself on the circular conducting plane A when acted upon by fluid distributed in any way in its own plane.
For this purpose, let us in the first place .conceive a quantity q of fluid concentrated in a point P, where r=a and 0 = 0, to act upon a conducting plate whose radius is unity. Then the value of V due to this fluid will evidently be
and consequently the equation of equilibrium analogous to the one marked (20) Art. 10, will be
(a? – 2ar cos 0 + my tot V................ (27);
V being due to the fluid on the conducting plate only.
If now we expand the value of V deduced from this equation, and then compare it with the formulæ (25) of the present article, we shall have generally E" = 0, and
except when i = 0, in which case we must take only half the quantity furnished by this expression in order to have the core rect value of C. Hence whatever u may be,
2 sin ( 17) efn = 0, and cer=-
the particular value i=0 being excepted, for in this case we have agreeably to the preceding remark
and then the only remaining exception is that due to the constant quantity on the left side of the equation (27). But it will be more simple to avoid considering this last exception here, and to afterwards add to the final result the term which arises from the constant quantity thus neglected.
The equation (26) of the present article gives by substituting for en hy its value just found,
the particular value Al being one half only of what would result from making i=0 in this general formula.
But everyone = 0) evidently gives EM = , and therefore the expansion of f (ac', y) before given becomes
f (9', y') = A + A cos O + Al* cos 26+ Al cos 30*+&c.
or by summing the series included between the braces,
A being the distance between P, the point in which the quantity of Auid q is concentrated, and that to which the density p is supposed to belong.
Having thus the value of f (x, y) we thence deduce
The value of p here given being expressed in quantities perfectly independent of the situation of the axis from which the angle & is measured, is evidently applicable when the point P is not situated upon this axis, and in order to have the complete value of p, it will now only be requisite to add the term due to the arbitrary constant quantity on the left side of the equation (26), aud as it is clear from what has preceded, that the term in question is of the form
const. x (1 -mg we shall therefore have generally, wherever P may be placed,
The transition from this particular case to the more general one, originally proposed is almost immediate: for if p represents the density of the inducing fluid on any element do, of the plane coinciding with that of the plate, p do, will be the quantity of fluid contained in this element, and the density induced thereby will be had from the last formula, by changing q into e,do. If then we integrate the expression thus obtained, and extend the integral over all the fluid acting on the plate, we shall have for the required value of p
R being the distance of the element do, from the point to which p belongs, and a the distance between do, and the center of the conducting plate.
Hitherto the radius of the circular plate has been taken as the unit of distance, but if we employ any other unit, and sup
pose that b is the measure of the same radius, in this case we shall only have to write my and in the place of a, s', do, and I respectively, recollecting that is a quantity of the dimension O with regard to space, by so doing the resulting value of p is
By supposing n = 2, the preceding investigation will be applicable to the electric fluid, and the value of the density induced upon an infinitely thin conducting plate by the action of a quantity of this Auid, distributed in any way at will in the plane of the plate itself will be immediately given. In fact, when n = 2, the foregoing value of p becomes
If we suppose the plate free from all extraneous action, we shall simply have to make e ll in the preceding formula; and thus
Biot (Traité de Physique, Tom, II. p. 277), has related the results of some experiments made by Coulomb on the distribution of the electric fuid when in equilibrium upon a plate of copper 10 inches in diameter, but of which the thickness is not specified. If we conceive this thickness to be very small compared with the diameter of the plate, which was undoubtedly the case, the formula just found ought to be applicable to it; provided we except those parts of the plate which are in the immediate vicinity of its exterior edge. As the comparison of any results mathematically deduced from the received theory of electricity with those of the experiments of go accurate an observer as Coulomb must always be interesting, we will here give