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 7't 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

(M– mS) (G−g)_gD, or to M

Mg
G

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 84 or from 84 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 ficient fluid will be lodged on the surfaces aß and 84, AB and DF will be not much over or undercharged.

redundant and de

and the coatings 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 Ve 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, Rr.Ss, 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.

174] COR. The redundant and deficient fluid in the intermediate spaces will in reality be not exactly equal and similarly disposed to that in DF, and in all probability the quantity of deficient fluid disposed near the extremity of DF will be greater than that in the corresponding parts of Pp, Ss, &c., or than the redundant fluid in the corresponding parts of Rr, Tt, &c., so that the redundant fluid in AB will perhaps be disposed rather less uniformly than it would be if the deficient and redundant fluid in those spaces was equal to and similarly disposed to that in DF; but on the whole there seems no reason to think that it will be much less, if at all less, uniformly disposed than it would be if the thickness of the glass was equal to the sum of the thicknesses of the spaces in which the fluid is immoveable, and the whole fluid within the glass was immoveable.

APPENDIX.

175] As the following propositions are not so necessary towards understanding the experiment as the former, I chose to place them here by way of appendix.

PROP. I. Let everything be as in Prop. XXXIV., except that the bodies H and L are not required to be at an infinite distance from the plates of glass; let now an overcharged body N be placed near the glass in such manner that the force with which it repels the column CG towards G shall be to that with which it repels the column EM towards M as the force with which the deficient fluid in DF attracts the column CG is to that with which it attracts EM: it will make no alteration in the quantity of redundant fluid in AB, provided the repulsion of N makes no alteration in the manner in which the fluid is disposed in each plate.

For increase the deficience of fluid in DF so much as that that coating and N together shall exert the same attraction on EM as DF alone did before, they will also exert the same attraction on CG as DF alone did before, and consequently the fluid in the two canals will be in equilibrio.

176] COR. In like manner, if the forces with which the body N repels the columns CG and EM bear the same proportion to each other as those with which the plate AB repels those columns, and therefore bear very nearly the same proportion to each other as those with which EM repels those columns, the quantity of deficient fluid in DF will be just the same as before N was brought near, and the redundant fluid in AB will be diminished by a quantity whose repulsion on C'G is the same as that of N thereon.

Therefore, if the repulsion of N on CG is not greater than that of H thereon, the diminution of the quantity of redundant fluid in AB will bear but a very small proportion to the whole. For the quantity of redundant fluid in AB is many times greater than that which would be contained in it if DF was away, id est, than that whose repulsion on CG is equal to the repulsion of H thereon in the contrary direction.

177] PROP. II. From the preceding proposition and corollary we may conclude that if the force with which N repels the columns CG and EM bears very nearly the same proportion to each other as the force with which DF attracts those columns, the quantity of redundant fluid in AB will be altered by a quantity which will bear but a very small proportion to the whole, unless the repulsion of N on CG is much greater than that of H thereon.

If the reader wishes to see a stricter demonstration of this proposition, as well as to see it applied to the case in which the fluid is supposed moveable in the intermediate spaces, as in Prop. XXXV., he may read the following:

=

178] PART 1. Take Ee thickness of those spaces in which the fluid is moveable, draw def equal and similar to DEF, and let the deficient fluid therein be equal to that in DF: the repulsion of the intermediate spaces on EM is to the difference of the attractions of DF on M and eu (supposing Ee and Mu to be equal to CE) very nearly as twice Ee to CE, and is therefore very nearly equal to twice the difference of the attraction of df and DF on EM.

In like manner the attraction of the intermediate spaces on CG is very nearly equal to twice the difference of the attraction of DF and df thereon.

Suppose now the quantity of deficient fluid in DF to be increased in the ratio of 1+f to 1, the redundant fluid in AB remaining the same as before, a new attraction is produced on EM, very nearly equal to

f 2

ƒ× (attraction of DF on EM) - × 2 (diff. attr. of dƒ and DF on EM),

that is, very nearly equal to ƒ× (attraction of df on EM).