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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 L 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 H 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

185]

INDUCTION ON COATED PLATE.

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manner as to repel EM with very nearly the same force that it does CG 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.,

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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

of the canal CG to be such that the repulsion of the coatings AB and DF thereon shall be not sensibly less than if it was infinite, and that the thickness of the glass Gg is very small in respect of the distance of R from it, and that the repulsion of R does not sensibly alter the disposition of the fluid in Tt and Ss, and also that the repulsion of R on GC and EM together is not much less than if GM was infinite, and also not much greater than the repulsion of the glass NnvV on CG.

For let the quantity of fluid in Tt and Ss be so much altered that the united repulsion of R and those two coatings on the two canals GC and EM together, and also their repulsion on gP, shall be the same as that of the two coatings alone before the approach of R.

By Prop. II. Cor. 1. the quantity of fluid in Tt will be very little altered thereby, for the repulsion of R on the canal gP is very nearly the same as its repulsion on gC and EM together.

As the repulsion of Tt, Ss and R together on the two canals GC and EM together is the same as before the approach of R, it follows that if their repulsion on gC is less than before, their repulsion on EM will be as much increased.

Let now the quantity of fluid in AB and DF be so much altered that their repulsion on gC shall be as much diminished as that of Kkil and R on the same column is diminished, and that their repulsion on EM shall be as much diminished as that of Kkil and R on the same is increased, it is plain that the fluid in all three canals will be exactly in equilibrio, and by the preceding corollary the quantity of fluid in AB will be very little altered, and therefore the proportion of the redundant fluid in AB and Tt to each other will be very little altered*.

186] COR. VII. By Prop. [XXIV. Art. 86] all which is said in this proposition and corollaries holds good equally whether the canals GC, EM and GP are straight or crooked.

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187] Let DE be an uniform canal of incompressible fluid infinitely continued towards E, and let A and B be given points in a right line with D, and let AB be bisected in C, the force with which any particle of fluid repels this canal (supposing the repulsion to be inversely as the square of the distance) is inversely as its distance from the point D, and therefore the sum of the forces with which two equal particles of fluid placed in A and B repels this canal is to the sum of the forces with which they would repel it if both collected in the point C,

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188] Let us now examine how far the proportion of the quantity of fluid in the large circle and the two small ones in Experiment v., [Art. 273] Fig. 18, bear to each other will be affected by the circumstances mentioned in [Art. 276], supposing the plates to be connected by canals of incompressible fluid.

First it appears from Cor. [VII. Art. 186], that the quantity of redundant fluid in the large circle, and also in the two small ones, will bear very nearly the same proportion to that in the jar A as it would if it had been placed at an infinite distance from A, for the distance of the plate from the jar was in neither experiment less than 63 inches, and neither the length nor the diameter of the coated part of the jar exceeded four inches, so that the repulsion of the jar on the canal connecting it to the plate could not differ by more than part from what it would be if the canal was infinitely continued, and would most probably differ from it by not more than or part of that quantity*; for the same reason the deficience of fluid in the trial plate will bear very nearly the same proportion to that in the jar, &c. as it would if it had been placed at an infinite distance from it.

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*The repulsion of a globe 4 inches diameter on a straight uniform canal of incompressible fluid extending 63 inches from it differs by only part from what it would be if the canal was infinitely continued, but the repulsion of a Leyden vial of that size on the same column differs probably not more than or of that quantity from what it would be if infinitely continued.

It is plain that if the plates had been placed at such a distance from the jar that the quantity of fluid in them had been considerably less than if they had been placed at an infinite distance, still the quantity in the large circle would bear very nearly the same proportion to that in the two small ones as it would if they had been placed at an infinite distance.

189] Secondly, it is plain that in trying the large circle, the repulsion of that circle increases the deficience of fluid in the trial plate, and the attraction of the trial plate increases the redundance in the circle. Now the repulsion of the plate Ee on the canal mM Na, and the attraction of the trial plate T on TRSA (supposing mMNa and rRSA to be infinitely continued beyond a and A) are by [Cor. IV. Art. 183] very nearly the same as if the redundant fluid in Ee and the deficient fluid in T were both collected in the centers of their respective plates, and the quantity of redundant fluid in Ee may be considered as equal to the deficient in T, and consequently the repulsion of Ee on m MNa is very nearly equal to the attraction of T on rRSA. Moreover, the repulsion of Ee on its own canal rRSA must be equal to the attraction of T on mMNa, as the jars with which they communicate are both equally electrified, and therefore, by Cor. [IV.], the quantity of redundant fluid in Ee will be increased in very nearly the same ratio as the deficient in T.

190] In like manner, in trying the two small circles, the quantity of redundant fluid in them is increased in very nearly the same ratio as the deficient in T, for as half the distance of the two circles never bore a greater proportion to em than that of 18 to 72, the repulsion of the two circles on the canal mM Na will be very nearly the same, and the deficience of fluid in 7 will be increased in very nearly the same ratio as if all the redundant fluid in them were collected in e, the middle point between them.

The quantity of redundant fluid in Bb indeed will be increased in a rather greater ratio, and that in Cc in a rather less ratio than if it was placed at, but the ratio in which the quantity of fluid in Bb is increased must very nearly as much exceed that in which it would be increased if it was placed at eas that in which Cc is increased falls short of it, as the attraction of T on the canal ƒRSA exceeds that on TRSA by nearly the same quantity as its attraction gRSA falls short of it, and therefore the quantity of redundant fluid in both circles together is increased in very nearly the same proportion as that in a circle placed in would be, and consequently the redundance in the two circles is increased in very nearly the same ratio as the deficience in the trial plate*.

* Memorandum relating to the second article.

191] The attraction of the trial plate on the canals fRSA and gRSA and the repulsion of the circles Bb and Cc on the canal mMNa is very nearly the same as if the deficient or redundant fluid in the plates was collected in the centre of their respective plates, and therefore the repulsion of the circles Bb and Ce on the canal mM Nn is inversely as the distances of their centres from m, and the increase of the quantity of redundant fluid in the circles Bb and Ce by the attraction of T is in the same proportion.

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