CN CN MR3 CR3 differs very little from is very small in respect of MC and as MC MC CN CN or the dif from MC MP3 ference of the repulsions of R on the points M and C differs very little the difference of the repulsions of T on the same points. Secondly, if CR is not considerably greater than MC, CN must be very small in respect of CR, and consequently must be very small in CN CN respect of MC. Therefore is very small in respect of CR3 MR3 CN and therefore the difference of the repulsions of R on C and M MC MT3 MR39 159] COR. Therefore by the same method of reasoning as was used in Cor. to Lemma XVI., the difference of the repulsions of the whole plate ACB on the points M and C is very nearly the same as if each particle of matter in it was transferred to the plane Tt and placed at the same distance from C as before, and therefore its repulsion on M is very nearly equal to its repulsion on C, provided MC is very small in respect of the least distance of the circumference of the plate from C, and that the thickness of the plate is everywhere very nearly the same, except at such a distance from C as is very great in respect of MC. 160 PROP. XXXIV.* Fig. 8. Let NnvV be a plate of glass or any other substance which does not conduct electricity, of uniform thickness, either flat, or concave on one side and convex on the other, and let the electric fluid be unable to penetrate at all into the glass or to move within it. Let ACB and DEF be thin coatings of metal, or any substance which conducts electricity, applied to the glass. * This proposition is nearly the same as Prop. XXII., only made more general. Let these coatings be of any shape whatsoever, and let their edges correspond as in Lemma XVI. Cor. Î. Let AB communicate with the body II, and DF with the body L, by the straight canals CG and EM of incompressible fluid. Let the points C and E be so placed that the two canals shall form one right line perpendicular to AB at the point C, and let the lengths of these canals be so great that the repulsion of the coatings on the fluid in them shall be not sensibly less than if they were infinite, and let H be overcharged and let L be saturated. It is plain from Prop. XII. that DF will be undercharged, and that AB will be more overcharged than it would otherwise be. Let Ww be a thin flat circular plate whose center is C, perpendicular to CE, and whose area is equal to that of AB, let the force with which the redundant fluid in AB would repel the short column CE (if ME was continued to C) be called m, and let the force with which it would repel CM, or with which it repels CG (for they are both alike), be called M. Let the force with which the same quantity of redundant fluid disposed in DF, in the same manner in which the deficient fluid therein (CE) EG is actually disposed, would repel { be called, let the force with which the same quantity of redundant fluid uniformly disposed on Ww would repel CG be called W, and let the force with which H repels CG be the same with which a quantity of fluid, which we will call B, uniformly distributed on Ww would repel it in the contrary direction: GW then will the quantity of redundant fluid in AB be B × Mg + Gm-mg' which, if M and G are very nearly alike, and m and g are very small in BW respect of G, differs very little from and the deficient fluid in g+ m DF will be to the redundant fluid in AB as M m to G, and therefore on the same supposition will be very nearly equal to it. For the force with which AB repels the fluid in EM must be equal to that with which DF attracts it, for otherwise some fluid would run out of DF into L, or out of L into DF. For the same reason the excess of the repulsion of AB on CG above the attraction of DF thereon must be equal to the force with which a quantity of redundant fluid equal to B spread uniformly on Ww would repel it. By the supposition the force with which AB repels the canal EM is M-m, and the force with which the same quantity of redundant fluid, spread on DF in the same manner in which the deficient fluid therein is actually disposed, repels it is G, therefore if the redundant fluid in AB is called 4, the deficient fluid in DF will be A × M M-m G m ; therefore and the excess " G the force with which DF attracts CG is (G-g) of the force with which AB repels CG above that with which DF attracts it is which must be equal to the force with which a quantity of fluid equal to B spread uniformly over Ww would repel it, that is, it must be equal to W B Α ; therefore A equals BGW 161] COR. I. If the plate of glass is flat, and its thickness is very small in respect of the least distance of the point C from the circumference of AB, and the fluid in AB and DF is spread uniformly, the Bx CW quantity of redundant fluid in DF will differ very little from 20E and the deficient fluid in DF will be very nearly equal to the redundant fluid in AB. For as the plate of glass is flat, the two coatings will be equal to each other, and therefore M and G are equal to each other, and so are g m and 9, and differs and moreover " W small in respect of G. little from very CE* is g very 162] COR. II. If the plate is flat and the two coatings are circular, their centers being in C and E, will be more accurately equal to the quantity of redundant fluid in AB B× CW CW CW being in this case equal to the semi-diameter of the coatings, and the deficient fluid in DF will be to the redundant in AB nearly as CW - CE to CW. which, if CE is small in respect of CW, differs very little from 2CE (CW - CE) 163] COR. III. If the plate of glass is not flat, and its thickness is very small in respect of the radius of curvature of its surface at and near C, everything else being as in Cor. I., the quantity of B× CW redundant fluid in AB will still be very nearly equal to 2CE For as CE is very small in respect of the radius of curvature, the two coatings will be very nearly of the same size, and therefore G differs very little from M, and m+g is to W very nearly as CE to CW*, and moreover m and g are both very small in respect of M and G†. 164] COR. IV. If we now suppose that the density of the redundant fluid in AB is greater at its circumference than it is near the point C, and that its density at and near C is less than the mean density, or the density which it would everywhere be of if it was spread uniformly, in the ratio of 8 to one, and that the deficient fluid in DF is spread nearly in the same manner as the redundant in AB, the quantity of redundant fluid in AB will be greater than before in a ratio approaching much nearer that of one to 8 than to that of equality, and that whether the glass is flat or otherwise. For by Lemma [XVI. Cor. II.], m and g will each be less than before in the above-mentioned ratio. 165] COR. V. Whether the plate of glass is flat or concave, or whatever shape the coatings are of, or whatever shape the canals CG and EM are of, or in whatever part they meet the coatings, provided the thickness of the plate is very small in respect of the smallest diameter of the coatings, and is also sufficiently small in respect of the radius of curvature of its surface in case it is concave, the quantity Bx CW of redundant fluid in AB will differ very little from 2CE For suppose that the canal GC meets the coating AB in the middle of its shortest diameter, and that the point in which ME meets DF is opposite to L, as in Prop. [XXII. Art. 74], the thickness of the glass will then be very small in respect of the distance of the point C from the nearest part of the circumference of AB, and moreover, by just the same reasoning as was used in the Remarks to Prop. XXII., it may be shown that 8 will in all probability differ very little from one, and consequently by Cors. I. and III. the redundant fluid in AB will be as above assigned. But by Prop. XXIV. the quantity of fluid in the coatings will be just the same in whatever part the canals meet them, or whatever shape the canals are of. * Lemma XVI. Cor. + As the demonstration of the sixteenth Lemma and its corollary is rather intricate, I chose to consider the case of the flat plate of glass separately in Cor. I., and to demonstrate it by means of Lemma XV. 166] COR. VI. On the same supposition, if the body H is a globe whose diameter equals Ww, id est the diameter of a circle whose area equals that of the coating AB, the redundant fluid in AB will be to that in H very nearly as CW to 4CE. For the quantity of redundant fluid in H will be 2B. 167] COR. VII. On the same supposition the redundant fluid in AB will be very nearly the same whether the glass is flat or otherwise, or whatever shape the coatings are of. 168] COR. VIII. On the same supposition, if the size and shape of the coatings and also the thickness of the glass is varied, the size and quantity of redundant fluid in H remaining the same, the quantity of redundant fluid in AB will be very nearly directly as its surface, and inversely as the thickness of the glass. 169] PROP. XXXV. (Fig. 9). Let Pp, Rr, Ss, Tt represent any number of surfaces whose distance from Nn, and consequently from Fig. 9. each other, is the same in all parts, and let everything be as in the preceding proposition, except that the fluid in the spaces PprR, SstT, &c., that is, in the spaces comprehended between the surfaces Pp and Rr, and between Ss and Tt, &c. is moveable*, in such manner, however, that though the fluid in any of these spaces as PprR is able to move freely from Pp to Rr or from Rr to Pp, in a direction perpendicular to the surface Pp or Rr, yet it is not able to move sideways, or in a direction parallel to those surfaces+, and let the fluid in the remaining spaces NnpP, RrsS, Ttv V, &c. be immoveable: the quantity of redundant fluid in AB and the deficient fluid in DF will be very nearly the same that they would be if the whole fluid within the glass was immoveable, and its thickness was only equal to NP+RS+ TV, &c., that is, to the sum of the thicknesses of those spaces in which the fluid is immoveable, provided that NV, the thickness of the glass, is very small * To avoid confusion I have drawn in the figure only two spaces in which the fluid is supposed to be moveable, but the case would be just the same if there were ever so many. + [Note 15.] |