The contrary would happen, according to our for mula, if the quantities e,e of electricity introduced into the two spheres were of a different nature, for then it would be necessary in the calculation to give to them different signs. The formula (1), which re presents the total result of the pressures exerted against the air parallel to the line of the centres; will then become positive, that is to say, according to what has been already agreed on, the external air will be more pressed in the direction AB, than in that of BA ; the sphere A will move then in the di rection in which the external pressure will have be come the weakest, that is towards the sphere B, agreeably to observation.
We have hitherto only considered the effect of the pressures round the sphere A, but the same reason ings and calculations will apply to the other sphere; only we must then employ, instead of E and E the expressions of the electric strata which correspond to them, and which we have seen to be ' E = Ser2 cos. u' 5er2r — 1 —.3 cos.' u'). It might be shown for this case, as well as for the other, that there cannot be any inequality of pressure but in the direction of the co-ordinates a; then comparing the points of the surface which correspond to the two sides of the plane of the co-ordinates y z, we shall find that element of the resulting force of the pressures parallel to the co-ordinates x, is in general Kra/ expressed by (E'' — in representing by E' what E' becomes when we change + a into — thSt is to say, + cos. u' into — cos. u', the angle 14' being here reckoned from the point a, situated on the side of A upon the line of the centres. By next considering the spheres as very distant we will ob tain in the same manner the value of E' — E:. Ap proximating no farther than the first power of this will give E' — E: = — cos. u'. We have then a' take E E' = 2 e and putting these va- lues in the expressions of the partial resulting force, C08. 9 will become . It may be de- , .
monstrated as above, that the sum of the factors is' will be proportional to the square of the radius of the sphere B, and may besides be repre sented by K'r sphere K' being the same numerical co.
efficient we have already employed. For the total resulting farce then, the expression will finally be -12K come that is, exactly the same which we have obtained for the other sphere, which ought to be the case, since in these sort of phenomena action and reaction are always equal. Here, as in the example immediately above, the positive sign of the expression will signify that the resulting force of the pressures exerted against the air round the sphere B, is directed to wards the other sphere, and the negative sign will signify that this resulting force is directed the oppo site way. The first case will take place when the electric charges ee are of a contrary nature ; in that case, the sphere B will advance from the side where the atmospheric pressure is weakest, that is towards A; the other case will happen when the electric charges ee are of the same nature, then B will re cede from A.
The common expression for the result of the pressures vanishes for both the spheres, when e or e is nothing, that is, when one of them is in the natu ral state. This seems to indicate that they would then neither approach nor recede from each other, while, in reality, we know that in this case they always ap proach. This apparent contradiction is owing to the degree of approximation at which we stopped our developement of the above expression. We have supposed our two spheres very distant from each other, compared with the radii of tileir surfaces; the result of this is, that whatever be the quantity of external electricity which we have introduced into each of them, it will distribute itself almost uniformly over the two hemispheres, anterior and posterior; so that the difference of the pressures exerted against the air by these two hemispheres, which is the only cause of motion, will be very small, and it is to this degree of minuteness that we have confined our approximations in developing V — If, however, the one of the two spheres, B for ex ample, is only electrified by the influence of the other, which we always suppose very distant, the developement of its natural electricities will be still very feeble, and of the same order of minuteness with that to which we have confined our approxima tions; but this weak electricity still dividing itself between the two hemispheres of B, in a manner near ly equal, as in the example immediately above, the difference of pressures round the two hemispheres will become very minute in a still lower degree—will become a quantity of the second order of minute ness, and, consequently, cannot be found in our de velopements, such as we have limited them. To ob tain it complete, we must not, in the calculation of E' E: confine ourselves to quantities, independent of - , but take its whole value. We will then have, a first of all, Se 5e r" r' E' = cos, u' (1 — 3 cos.' u') a2 2 then changing + a into — x, or + cos. te into ? cos. u, we will have, This complete value of E' E: will now no more vanish when e' is nothing, but it will be seen that the ,terms which remain are of the order of those which we have neglected in our first approximation. Making, then, here e' equal to nothing, it remains, becomes ti (I u').
a It only remains to take the sum of it over all the extent of the surface of the sphere B ; but, in this operation, the variable fhctor, •,' u' (I — u'), will give a result proportional to the square of the radius r' of the sphere B, and which we may consequently represent by K" K" being a constant numerical co-efficient different from K'; thus, the total resulting force will at last be