which, if CE is small in respect of CW, differs very little from

2CE (CW-CE)
ᏟᎳ -

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 Bx 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 Gt.

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 I 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.]

in respect of the smallest diameter of AB, and also in respect of the radius of curvature of the surface of the glass.

Let the canals GC and EM be perpendicular to the plate of glass and opposite to each other, so as to form one right line, and let them meet AB and DF in the middle of their shortest diameters. The coating AB will be very much overcharged, and DF almost as much undercharged, in consequence of which some fluid will be driven from the surface Pp to Rr and from Ss to Tt. Moreover the quantity of fluid driven from any portion of the surface Pp near the line CE will be very nearly equal to the quantity of redundant fluid lodged in the corresponding part of AB, or more properly will be very nearly equal to a mean between that and the quantity of deficient fluid in the corresponding part of DF.

For a particle of fluid placed anywhere in the space PprR near the line CE is impelled from Pp to Rr by the repulsion of AB and the attraction of DF, and it is not sensibly impelled either way by the spaces SstT, &c., as the attraction of the redundant matter in Ss is very nearly equal to the repulsion of the redundant fluid in Tt; and moreover the repulsion of AB on the particle and the attraction of DT are very nearly as great as if their distance from it was no greater than that of Pp and Rr, and therefore the particle could not be in equilibrio unless the quantity of fluid driven from Pp to Rr was such as we have a signed.

As to the quantity of fluid driven from Pp to Rr at a great distance from CE, it is hardly worth considering. It is plain, too, that the quantity of fluid driven from Ss to Tt will be very nearly the same as that driven from Pp to Rr.

Let now G, g, M, m and W signify the same things as in the preceding proposition, and let the quantity of redundant fluid in AB be called A as before, and let NP + RS + TV + &c., id est, the sum of the thicknesses of those spaces in which the fluid is immoveable, be to NV, or the whole thickness of the glass as S to 1, and let PR + ST+ &c., or the sum of the thicknesses of those spaces in which the fluid is moveable be to NV as D to one.

Take EII equal to PR, the repulsion of the space PprR on the infinite column EM is equal to the repulsion of the redundant fluid in Rr on EII, and therefore is to the repulsion of AB on CE very nearly as EII or PR to CE. Therefore the repulsion of all the spaces Pprk, SstT, &c. on EM is to the repulsion of AB on CE very nearly as D to one, or is equal to mD, and therefore the sum of the repulsions of AB and those spaces together on EM is very nearly equal to M −m+mD or to M-mS.

But the attraction of DF on EM must be equal to the abovementioned sum of the repulsions, and therefore the deficient fluid in DF A (M-MS) must be very nearly equal to G

By the same way of reasoning it appears that the force with which

CG is repelled by AB, DF, and the spaces PprR and SstT, &c. together is very nearly equal to

mgS
G

which, as M differs very little from G, and is very small in respect

of mS or gS, is very nearly equal to g+mS-gD or to (g+m) S, therefore the quantity of redundant fluid in AB will be very nearly equal to

BW

and will therefore be greater than if the fluid within the glass (g+m) S' was immoveable very nearly in the ratio of one to S, or will be very nearly the same as if the thickness of the glass was equal to CE× S, and the fluid within it was immoveable.

170] PROP. XXXVI. Fig. 10. Let every thing be as in the preceding proposition, except that the electric fluid is able to penetrate into the glass on the side Nn as far as to the surface Kk, and on the side Vv

as far as to Yy; in such manner, however, that though the fluid can move freely from AB to aß or from aẞ to AB, and also from DF to dø or from do to DF, in a direction perpendicular to those surfaces, yet it is unable to move sideways, or in a direction parallel to those surfaces: the quantity of redundant fluid on one side of the glass, and of deficient fluid on the other, will be very nearly the same as if the spaces NnkK and VoyY were taken away and the coatings AB and DF were applied to the surfaces Kk and Yy.

For by [Art. 132] of former Part, almost all the redundant and deficient fluid will be lodged on the surfaces aẞ and 84, and the coatings AB and DF will be not much over or undercharged. Now if the whole

of the redundant and deficient fluid was lodged in aß and 84, it is evident that the quantity of redundant and deficient fluid would be exactly the same as if the spaces NnkK and VoyY were taken away, and therefore it will in reality be very nearly the same.

171] COR. I. Therefore the quantity of redundant fluid on the positive side of the glass, that is, in the coating AB, and the space AaßB together, as well as the quantity of deficient fluid on the negative side of the glass, will be very nearly the same that they would be if the fluid was unable to penetrate into the glass or move within it, and that the thickness of the glass was equal only to the sum of the thicknesses of those spaces in which the fluid is immoveable.

172] COR. II. Whether the electric fluid penetrates into the glass or not, it is evident that the quantity of redundant fluid on one side the glass, and of deficient fluid on the other, will be very nearly the same, whether the coatings are thick or thin.

173] PROP. XXXVII. It was shewn in the remarks on Prop. XXII. in the first Part, that when the plate of glass is flat, and the fluid within it is immoveable, the attraction of the deficient fluid in DF makes the redundant fluid in AB to be disposed more uniformly than it would otherwise be. Now if we suppose the fluid within the glass to be moveable as in the preceding proposition, and that the deficient fluid in the planes Pp, S8, &c. and the redundant fluid in the planes Rr, Tt, &c. is equal to, and disposed similarly to that in DF, the redundant fluid in AB will be disposed more uniformly than it would be if the fluid within the glass was immoveable, and its thickness no greater than the sum of the thicknesses of those spaces in which the fluid is immoveable.

For let the intermediate spaces be moved so that Tt shall coincide with Vo and Rr with Ss, &c., but let the distance between Tt and Ss and between Rr and Pp, &c. remain the same as before, that is, let the thickness of the spaces in which the fluid is moveable remain unaltered. The distance of Pp from Nn will now be equal to the sum of the thicknesses of the spaces TtVv, RrSs, NnPp, &c. in which the fluid is immoveable.

Now, after this removal, the effect of the planes Tt and DF and of Rr and Ss, &c. will destroy each other, so that the intermediate spaces and DF together will have just the same effect in rendering the redundant fluid in AB more uniform than the plane Pp alone will have, that is, the fluid in AB will be disposed in just the same manner as if the thickness of the glass was no greater than the sum of the thicknesses of the spaces in which the fluid is immoveable, and the whole fluid within the glass was immoveable.

But the effect of the intermediate spaces in making the fluid in AB more uniform was greater before their removal than after, for the effect of the two planes Pp and Rr together, and also that of Ss and Tt together, &c. is the greater the nearer they are to AB.