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 CG is the same as that of N thereon.

Therefore, if the repulsion of N on CG is not greater than that of II 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

ƒ× (attraction of DF on EM)

f

x 2 (diff. attr. of df and DF on EM),

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

In like manner a new attraction is produced on CG, very nearly equal to fx (attraction of df on CG), therefore, the new attraction produced on EM is to that produced on CG very nearly as the attraction of df on EM is to its attraction on C'G, and therefore in order that the quantity of redundant fluid in AB shall not be altered by the approach of N, the repulsion of N on EM must be to its repulsion on CG very nearly as the attraction of df on EM to its attraction on CG.

179] PART 2. Let the fluid within the glass be either moveable, as in Prop. [XXXV. Art. 169], or let it be immoveable, and let the distance of H and L from the glass be either great or not.

Let the repulsion of H on sum of these repulsions = S.

GC
EM

in direction GC be!!

and let the

(GC in direction CG

=

Α

EM in direction EM = B

Hp'

and let the re

Let the repulsion of N on pulsion which N should exert on CG in order that the redundant fluid in AB should remain unaltered be to that which it should exert on EM:: 1: P.

The quantity of redundant fluid in AB will be increased in the B-PA 1+p ratio of 1+ to 1, which, if P differs very little from 1,

differs very

S P+P little from that of

but the latter part of these two repulsions, or the force

[ PA + B
P+p
PPA + PB
P+P

has no tendency to alter the redundant fluid in AB, but the first part, or the force

180] COR. I. If the lengths of the columns CG and EM are such that the repulsion and attraction of AB and DF on them are not sensibly less than if they were of an infinite length, the attraction of DF on CG will be very nearly equal to its attraction on EM, and therefore, if the forces with which N repels the columns CG and EM are very nearly equal to each other, the quantity of redundant fluid in AB will be very little altered thereby.

N.B. If the size of H is much greater than that of AB, it is possible that its distance from the glass may be such as to exert a very considerable repulsion on EM, and yet that the action of AB and DF on CG shall be not sensibly less than if it was of an [infinite length].

181] COR. II. Let the bodies H and L be of the same size and shape and at an infinite distance from the glass, and let the fluid be in equilibrio. Let now an equal quantity of fluid be taken from H and I, the quantity of redundant fluid in AB will be very little altered thereby.

For the repulsion of the whole quantity of fluid in Z on the canal EM will be as much diminished as that of H on CG, so that it comes to the same thing as placing an overcharged body N in such manner that its repulsion on CG shall be equal to that on EM, which by the preceding proposition will make very little alteration in the quantity of redundant fluid in AB.

182] COR. III. Let the bodies I and I be at an infinite distance, and either of the same or different size, and let the fluid be in equilibrio. Let now the body H be brought so near to AB that its repulsion on GC shall be sensibly less than before. The quantity of redundant fluid in AB will be very little altered thereby, provided the repulsion of the two plates on the column CG is not sensibly diminished.

For whereas when I was at an infinite distance from AB it exerted no repulsion on EM, now it is brought nearer it does exert some, and its repulsion on EM is very nearly equal to the diminution of its repulsion on CG, so that it comes to the same thing as placing a body N in such

manner as to repel EM with very nearly the same force that it does C'G in the contrary direction.

183] COR. IV. Let the body H be brought near AB as in the preceding corollary, and let the fluid be in equilibrio; let now an overcharged body R be placed near H, the quantity of redundant fluid in H must be so much diminished, in order that the fluid may remain in equilibrio, supposing the fluid in AB to remain unaltered, as that the diminution of its repulsion on the two columns GC and EM shall be equal to the repulsion of R on the same columns. Consequently, if the repulsion of R on them is to the repulsion which H exerted on them before the approach of R as n to 1, the quantity of redundant fluid in H will be diminished in the ratio of 1-n to 1.

For supposing the quantity of fluid in H to be thus diminished, I say, the quantity of fluid in A will remain very nearly the same as before. For the repulsion of H and R on the two columns will be the same as that of H was before, but it is possible that their repulsion on GC may be a little less, and their repulsion on EM as much greater than that of H was before, but this, by the preceding corollaries, will make very little alteration in the quantity of fluid in AB.

184] COR. V. It appears from Prop. XXIII. that the repulsion of the body R on the two columns GC and EM will be the same in whatever direction it is placed in respect of H and the canal, provided its distance from the point G is given, and consequently the diminution of the quantity of fluid in the body H will be very nearly the same in whatever direction R is situated, provided its distance from G is given.

185] COR. VI. Fig. 11. Suppose now that instead of the body I there is placed a plate of glass Kkil, coated as in Props. XXXIV. and XXXV.,

with the plates Tt and Ss, whereof Tt communicates with AB by the canal GC, and the other Ss communicates by the canal gP with the body P, placed at an infinite distance and saturated with electricity, and let AB and consequently Tt be overcharged, and let the fluid be in equilibrio.

Suppose now that an overcharged body R is brought near the glass Kkil, I say that the proportion which the redundant fluid in Tt bears to that in AB will be very little altered thereby, supposing the length