For the repulsion of all the matter therein, when collected at C,

147] LEMMA XIV. The repulsion of CK on a particle at B, in the

CK

direction CB, is proportional to KB CB, supposing the base of CK and the size of the particle B to be given.

For supposing CK to flow, the fluxion of its repulsion on B in the

CK CB direction CB is proportional to X KB KB'

and is nothing when CK is nothing.

148] LEMMA XV. Fig. 3. bases are GEF and HMN and

the fluent of which is

CK KB x CB'

Let GEFHMN be a cylinder whose whose axis is CK. Let the convex

m

K

Fig. 3.

W

surface of this cylinder be uniformly coated with matter, and let GC be small in respect of CK. Let GA be a diameter of the base produced, and D any point therein. The repulsion of the convex surface of the cylinder on the point D in the direction CD is very nearly the same as if all the matter therein was collected in the axis CK and spread uniformly therein.

For let MED and med be two planes infinitely near to each other, parallel to CK and passing through D, and cutting the convex surface in ME and NF and in me and nf, which will consequently be right lines equal to each other and perpendicular to ED; and draw CP perpendicular to ED.

The repulsion of NnfF on D in the direction CD is proportional to Ff × FN PD

X and that of Mme E is proportional to
CD

Eex EM PD

X

EDxMD

CD'

Ff

Ee

FD

ED

But Ff is to Ee as FD to ED, therefore and are each equal

But the repulsion of the same quantity of matter collected in CK is

proportional to

1

1

(Fƒ+ Ee) × CK

X

2

2CD KD, and, as CG is small in respect

of CK, + differs very little from ND MD repulsions of MmeE and NnfF is very nearly the same as if all the matter in them was collected in CK, and consequently the repulsion of the whole convex surface of the cylinder will be very nearly the sam as if all the matter in it was collected in CK.

149] COR. Therefore if BA represents an infinitely thin cylindric column of uniform matter infinitely extended beyond A, the repulsion of the convex surface of the cylinder thereon in the direction BA is very

* As neither MD nor ND differ from KD by so much as CB, it is plain that 2 KD

1

+

1 MD ND

cannot differ from in so great a proportion as that of BC to KD, but

in reality it does not differ from it in so great a ratio as that of CB to KD2, but as it is not material being so exact, I shall omit the demonstration. See A. 1.

[From MS. "A. 1"] Demonstration of note at bottom of page 8, CB=r, CP=b, PF=d, PD=a, CR2+CD2=e2,

ND2 CR2+a2 - 2ad + d2 = e2 — b2 - 2ad + d2 = e2-f2 - 2ad

nearly the same as if all the matter therein was collected in CK, and therefore is to the repulsion of the same quantity of matter collected CK+ KB ᏟᏦ in the point C thereon very nearly as nat. log. to CB CB

that

2CK CK is very nearly as nat. log. to In like manner the repulsion on CB CB' the infinite column DA is to the repulsion of the same quantity of

matter collected in C very nearly as nat. log.

CK+ KD CK
to
CD'

CD

150] PROP. XXXI. Fig. 3. Let the cylinder GEFKMN be connected to the globe W, whose diameter is equal to GB and whose distance from it is infinite, by a canal TR of incompressible fluid of any shape, and meeting the cylinder in any part, and let them be overcharged the quantity of redundant fluid in the cylinder will be to that in the globe in a less ratio than that of CK to nat. log.

CK 2CB

CK

2CK

9 CB

and in

a greater ratio than that of to nat. log. provided CB is small

in respect of CK.

CB'

By Prop. XXIV. the quantity of redundant fluid in the cylinder will bear the same proportion to that in the globe in whatever part the canal meets the cylinder, therefore first I say the redundant fluid in the cylinder will bear a greater proportion to that in the globe than CK CK that of to nat. log. 2CB CB'

For let the canal TR be straight and perpendicular to BL, and let it meet the cylinder in R, the middle point of the line BL, and let it, if produced, meet the axis in S, which will consequently be the middle point of CK; then, if the redundant fluid in the cylinder was spread uniformly on its convex surface, the quantity of redundant fluid therein CK CK would be to that in the globe very nearly as to nat. log. 2CB CB

For in that case the repulsion of the cylinder on the canal RT would be to the repulsion of the same quantity of redundant fluid collected

2SK SK

CK CK
2CB'

in C very nearly as nat. log. to or as nat. log. CB to

SR SR

and

the force with which the globe repels the canal in the direction TR is the same with which a quantity of redundant fluid equal to that in the globe placed at S would repel it in the contrary direction.

But there can be no doubt but that almost all the redundant fluid in the cylinder will be collected on its surface, and also will be collected in greater quantity near the ends than near the middle, consequently the repulsion of the cylinder on RT will be less than if the redundant fluid was spread uniformly on its convex surface, and therefore the quantity of redundant fluid in it will bear a greater proportion to that in the globe than it would on that supposition.

Secondly, the quantity of fluid in the cylinder will bear a less proportion to that in the globe than that of to nat. log.

CK
CB

2CK
CB

For suppose the canal to meet the cylinder in B and to coincide with BA. Then, if the redundant fluid was spread uniformly on the convex surface, the quantity therein would be to that in the globe very

CK

nearly as to nat. log.

CB

2CK and the real quantity of redundant fluid CB' in it will bear a less proportion to that in the globe than if it was spread uniformly on the convex surface.

151] COR. Therefore the quantity of redundant fluid in the cylinder is to that in a globe whose diameter equals CK in a ratio between that of 2 to nat. log. and that of 1 to nat. log.

2CK
CB

CK*
CB

152] PROP. XXXII. Fig. 4. Let ADFB and adfb be two equal cylinders whose axes are EC and ec, let them be parallel to each other

D

Fig. 4.

M

B

a с

and placed so that Cc, the line joining the ends of the axes, shall be perpendicular to the axes, and let the lines EC and Fb be bisected in G and g, and let them be connected by canals of incompressible fluid of any shape to a third cylinder of the same size and shape placed at an infinite distance from them, and let them be overcharged: the quantity of redundant fluid in each of them will be to that in the third cylinder in EG+eg and that of

EC
CB

EC

a ratio between that of log to log + log

log

съ

CB

EC+Eb
съ

provided the redundant fluid in the

2EC 2EC to log + log CB CB third cylinder is disposed in the same manner as in the other two.

For let us suppose that ADFB and ad fb are connected to the third cylinder by the canal GM, then, if the redundant fluid in each cylinder is disposed uniformly on its convex surface, the sum of the repulsions of ADFB and adfb on the canal gM will be to the repulsion of the third

[* Note 12.]

cylinder thereon (supposing the quantity of redundant fluid in it to be 2EG EG+ Eg equal to that in each of the two others) as log + log

2EG

log CB

CB

Gg

to

Let us now suppose the fluid in the first two cylinders to be disposed so as to be in equilibrio, and consequently to be disposed in greater quantity near their extremities than near their middles, and let the fluid in the third cylinder be disposed in the same manner, and be the same in quantity as before. The repulsion of ADFB on Gg will be diminished in a greater ratio, and consequently its repulsion on gM will be diminished in a less ratio than that of adfb on gM, consequently the sum of the repulsions of ADFB and adfb on gM will be diminished in a less ratio than that of the third cylinder thereon, and therefore the sum of the repulsions of ADFB and adfb on gM will be to that of the third cylinder thereon in a greater ratio than that of

Therefore the real quantity of redundant fluid in each of the first two cylinders will be to that in the third cylinder in a less ratio than

EG+ Eg
Gg

In like manner, by supposing them to be connected to the third cylinder by the canal bD, it may be shown that the quantity of redundant fluid in either of the first two cylinders is to that in the third in a greater ratio than that of log

2EC
CB

2EC
CB

to log + log

EC + Eb*

Cb

153] PROP. XXXIII. If two bodies B and b are successively connected by canals of incompressible fluid to a third body C placed at an infinite distance from them, and are overcharged, that is, if one of them, as B, is first connected to C and afterwards B is removed and b put in its room, the quantity of redundant fluid in C being the same in both cases, it is plain that the quantity of redundant fluid in B will bear the same proportion to that in b that it would if B and b were placed at an infinite distance from each other, and connected by canals of incompressible fluid.

154] LEMMA XV. Fig. 5. Let AB be a thin flat plate of any shape whatsoever, of uniform thickness and composed of uniform matter. Let CG be an infinitely slender cylindric column of uniform matter perpendicular to the plane of AB and meeting it in C and extended infinitely beyond G. Let ab be a thin circular plate perpendicular to CG whose center is C. Let the area of ab be equal to that of AB, and let the quantity of matter in it be the same, and let it be disposed uniformly.

[* Note 13.]