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GENERAL PRELIMINARY RESULTS.
(1.) The function which represents the sum of all the electric particles acting on a given point divided by their respective distances from this point, has the property of giving, in a very simple form, the forces by which it is solicited, arising from the whole electrified mass.-We shall, in what follows, endeavour to discover some relations between this function, and the density of the electricity in the mass or masses producing it, and apply the relations thus obtained, to the theory of electricity.
Firstly, let us consider a body of any form whatever, through which the electricity is distributed according to any given law, and fixed there, and let c', y', z, be the rectangular co-ordinates of a particle of this body, p the density of the electricity in this particle, so that dac'dy'dz' being the volume of the particle, p'da'dy'dz' shall be the quantity of electricity it contains : moreover, let go be the distance between this particle and a point p exterior to the body, and V represent the sum of all the particles of electricity divided by their respective distances írom this point, whose co-ordinates are supposed to be x, y, z, then shall we have
the integral comprehending every particle in the electrified mass under consideration.
LAPLACE has shown, in his Méc. Céleste, that the function V has the property of satisfying the equation
o-dv dv. dy
and as this equation will be incessantly recurring in what follows, we shall write it in the abridged form () = 8V; the syan bol 8 being used in no other sense throughout the whole of this Essay.
In order to prove that 0=8V, we have only to remark, that by differentiation we immediately obtain () = 85, and consequently each element of V substituted for V in the above equation satisfies it; hence the whole integral (being considered as the sum of all these elements) will also satisfy it. This reasoning ceases to hold good when the point p is within the body, for then, the coefficients of some of the elements which enter into V becoming infinite, it does not therefore necessarily follow that V satisfies the equation
0 = 8V,
althongh each of its elements, considered separately, may do so.
In order to determine what & V becomes for any point within the body, conceive an exceedingly small sphere whose radius is a inclosing the point y at the distance b from its centre, a and b being exceedingly small quantities. Then, the value of V may be considered as composed of two parts, one due to the sphere itself, the other due to the whole mass exterior to it: but the last part evidently becomes equal to zero when substituted for V in 8V, we have therefore only to determine the value of 8 V for the small sphere itself, which value is known to be
S (2Ta'p - ***p); p being equal to the density within the sphere and consequently to the value of pat p. If now X, Y,, %, be the co-ordinates of the centre of the sphere, we have
7 = (x,- x)* + 14,- y) + (3,-2),
8 (2a p- bp) = -47p. Hence, throughout the interior of the mass
0=8V+47P; of which, the equation 0 = V for any point exterior to the body is a particular case, seeing that, here p=0.
Let now q be any line terminating in the point p, supposed without the body, then – G = the force tending to impel a particle of positive electricity in the direction of q, and tending to increase it. This is evident, because each of the elements of V substituted for V in- , will give the force arising from this element in the direction tending to increase q, and consequently, - will give the sum of all the forces due to every element of V, or the total force acting on p in the same direction. In order to show that this will still hold good, although the point p be within the body; conceive the value of V to be divided into two parts as before, and moreover let p be at the surface of the small sphere or b=a, then the force exerted by this small sphere will be expressed by
& trap Contato
da being the increment of the radius a, corresponding to the increment dg of g, which force evidently vanishes when a=0: we need therefore have regard only to the part due to the mass exterior to the sphere, and this is evidently equal to
v-a p. But as the first differentials of this quantity are the same as those of V when a is made to vanish, it is clear, that whether the point p be within or without the mass, the force acting upon it in the direction of q increasing, is always given by - ).
Although in what precedes we have spoken of one body only, the reasoning there employed is general, and will apply equally to a system of any number of bodies 'whatever, in those cases even, where there is a finite quantity of electricity spread over their surfaces, and it is evident that we shall have for a point p in the interior of any one of these bodies
0 = 8 + 47P .................... (1). Moreover, the force tending to increase a line q ending in any point p within or without the bodies, will be likewise given by
- ; the function V representing the sum of all the electric particles in the system divided by their respective distances from p. As this function, which gives in so simple a form the values of the forces by which a particle p of electricity, any how situated, is in pelled, will recur very frequently in what follows, we have ventured to call it the potential function belonging to the system, and it will evidently be a function of the co-ordiDates of the particle p under consideration.
(2.) It has been long known from experience, that whenever the electric fluid is in a state of equilibrium in any system whatever of perfectly conducting bodies, the whole of the electric fluid will be carried to the surface of those bodies, without the smallest portion of electricity remaining in their interior: but I do not know that this has ever been shown to be a necessary consequence of the law of electric repulsion, which is found to take place in nature. This however may be shown to be the case for every imaginable system of conducting bodies, and is an immediate consequence of what has preceded. For let x, y, &, be the rectangular co-ordinates of any particle p in the interior
Id of one of the bodies; then will – Go be the force with which p is impelled in the direction of the co-ordinate x, and lending to increase it. In the same way and he will be the forces in y and 2, and since the fluid is in equilibrium all these forces are equal to zero: hence
which equation being integrated gives
D = const. This value of V being substituted in the equation (1) of the preceding number gives
and consequently shows, that the density of the electricity at any point in the interior of any body in the system is equal to zero.
The same equation (1) will give the value of p the density of the electricity in the interior of any of the bodies, when there are not perfect conductors, provided we can' ascertain the value of the potential function V in their interior.
(3.) Before proceeding to make known some relations which exist between the density of the electric fluid at the surfaces of bodies, and the corresponding values of the potential functions within and without those surfaces, the electric fluid being confined to them alone, we shall in the first place, lay down a general theorem which will afterwards be very useful to us. This theorem may be thus enunciated:
Let U and V be two continuous functions of the rectangular co-ordinates x, y, z, whose differential co-efficients do not become infinite at any point within a solid body of any form whatever ; then will
the triple integrals extending over the whole interior of the body, and those relative to do, over its surface, of which do represents an element: dro being an infinitely small line perpendicular to the surface, and measured from this surface towards the interior of the body.