Electricity

equal in the points relatively to which the co-ordinate x is the same, and which are consequently situated upon one small circle, parallel to the plane of the co.. ordinates y, a. Besides, as all these points are equal ly distant from the line of the centres, it is clear that the total result of the equal forces which are applied to them will be in the direction of this line; conse quently this will also be the direction of the general result of all the efforts of this kind exerted upon the whole surface of the sphere A.

To obtain, now, easily, the sum of all these forces, parallel to the line of the centres, which is here that of the co-ordinates x, let us begin by joining together the values of x which are equal and contrary; for the thicknesses E, of the electric stratum on the two he mispheres of A being almost equal, from the suppo sition that the two spheres are very distant, the pres sures corresponding to opposite values x and — x, must be almost equal also; and, as the components which they give parallel to the co-ordinates x are in a contrary direction, their sum must be reduced to a very small quantity. To introduce this circumstance, call E what E becomes when we change -I- x into — x; then the expressions of the corresponding components parallel to the co-ordinates x will be, On the side of the) Tending to move positive oo-or.-I- K - the air in the di.

' dinates x, r rection AB On the side of the Tending to move negative co-or. x — K N the air in the di.

dinate.

We preserve the superficial element • always of the same value, because it is exactly alike in the two cases, on account of the ,symmetrical form of the sphere on the two sides of the plane of the co-ordi nates y z; adding these two components to each other with their actual sign, their sum will express the element of the total resulting force which tends to carry the air in the direction AB. This, then, only the first power of which, in the case we are considering, will contain an infinitely great propor tion of the total result, compared with the other powers, it will become dr' 2Sr Sr = C08. 1 — cos. ti ar a a cos or by reduction E—E. = . a.

This value of E—E must now be multiplied by E to form the factor 2 which enters into the expression of the total g force; but since E—E. is already of the order it is evident that, in E we must confine ourselves to the terms which are not divided by a; this limitation reduces the value of E to 2 e, and employing this to mul tiply there results consequently, by subtracting these equations from each other, we have E — r + 2 ar cos. u 1 cos. u 01) } or, what is the same thing, and is better adapted for approximations, E — E = 1 —; ) 1 (1 ar a 1 r2 1 u a Since we suppose the two spheres very distant from each other, compared with the magnitude of their radii, will be a very small fraction; hence we may develope this expression for E—E into a verging series of the ascending powers of , this will be effected by the binomial theorem; and taking

VOL. iv. PART I.

It only remains to substitute this value in the expres sion of the resulting force, parallel to the co-cadinatee x, which we have found equal toK x* for the superficial element *; and by putting for Each of these partial results is proportional to thecial element and to the square of the co angle, which ; elements form with the axis of the co-ordinates z. But, if we compare them together upon different spheres, this angle will always be expressed by the same values; for the equal va lues of u, however, the superficial element *will vary in magnitude proportionally to the square of the ra dius r of the sphere. Consequently, the sum of all the values of the factor • u, extended to every sphere, will only vary from each other in the ratio of the square it may be represented then by being a constant and numerical co-efficient which may be found, and which, in reality, is found by the processes of the integral calculus. Supposing it known then, the total result of the pressures parallel to the co-ordinates x will be (1).

It will be directly proportional then to the quantities 44rrle' of external electricity which they pos sess, and inversely proportional to the square of the distance of the two centres. When the quantities of electricity given tothe two spheres are of the same na ture, whether vitreous or resinous, the values of e and of e' must be considered as having the same sign. In that case the expression (1) is negative, that is to say, according to what has been previously admitted, that, in this case, the air which surrounds the sphere A, is more pressed in the direction BA, than in the direction AB. It will not then press equally the sphere A, as it did before it was electrified ; it will press it less on the side which is most distant from the other sphere, since it is in that direction that the electric reaction is the strongest. Consequent ly, if the sphere A is at liberty to move, and de prived of its weight, or if its weight be sustained by a thread of suspension, it will put itself in motion from the side where the atmospheric pressure has be come the weakest, that is to say, that it will recede from the other sphere B.