Let B be that point of the circumference of AB which is nearest to C. If EC is small in respect of CB, the repulsion of the plate AB on the short column EC is to the repulsion of ab on the infinite column CG nearly as EC to cb.

For let BD be a circle drawn through B with center C, as EC is very small in respect of CB, the repulsion of the circle BD on EC is to its repulsion on CG very nearly as EC to CB, and therefore is to the repulsion of ab on cG very nearly as EC to cb. But the repulsion. of AB on EC is very little greater than that of DB, for the repulsion of DB is very near as great as it would be if its size was infinite.

155] LEMMA XVI. Let ACB and DEF be two thin plates, not flat but concave on one side, let their distance be everywhere the same, Fig. 6. A

and let it be very small in respect of the radius of curvature of all parts of their surface. Let C be any point of the surface of AB, and let CE be perpendicular to the surface in that point. Let Tt be a flat plate perpendicular to CE.

Let R be any point in AB and S the corresponding point in DF, and let T be the corresponding point in Tt*: the sum of the repulsions of R on the column CE in the direction CE and of S on the same column in the opposite direction EC is very nearly equal to the force with which they would repel the same column in the direction CE if they were both transferred to T, provided CR is very small in respect of the square of the least radius of curvature of the surface of AB.

Let RS be continued till it meets CE continued in V, draw EM and SN perpendicular to CR.

Let CMC, RE - RM = E, SC - NCS, and SE - NM = D.

As CE is very small in respect of the least radius of curvature of AB, and CV is not less than the least radius of curvature, CM and NR. are each very small in respect of CR, and therefore CN, MR, and ES differ from CR in a very small ratio. Moreover as CR is very small in respect of CV, CM3 and RN are very small in respect of CE2, and therefore ME and NS differ in a very small ratio from CE; and, moreCE

over, 2 × (TE - TC) is greater than TE

Now the repulsion of the point R on the column CE in the direcᎡᎬ - ᎡᏟ

tion CE is

1

1 +

ᎡᏟ RE

==

RC RE, and the repulsion of the point S on

and the repulsion of the two particles when transferred to T on the column CE, or the repulsion of T, as I shall call it for shortness, is TE-TC

2

But as ME differs in a very small ratio from CE, and RM differs in a very small ratio from RC, RE - RM or E differs in a very small ratio from TETC. In like manner SC - NC or S differs in a very small ratio from TE - TC, and ER and CS both differ in a very small ratio from TE, and SE differs in a small ratio from TC.

* If RS is drawn perpendicular to the surface of AB at the point R cutting DF in S, I call S the corresponding point of the plate DF, and if CT is taken in the intersection of the plane RCE with that of the plate Tt equal to the right line CR, I call T the corresponding point of Tt.

+ Lemma XII. [Art. 146].

Moreover, as EM and SN differ very little from each other, D is very

each differ from one in a less ratio than that of RC RE

CE to CV, and therefore

differs from one in a less ratio than

SEX SC

Therefore the sum of the repulsions of R and S differs very little from the repulsion of T.

N.B. Though the distance CR is ever so great, it may be shown that the sum of the repulsions of R and S cannot be more than double that of T*.

1561 COR. I. Let the edges of the plates ACB and DEF correspond, that is, let them be such that if a line is erected on any part of the circumference of one plate perpendicular to the [tangent] plane of the plate in that part, that line shall meet the other plate in its circumference. Let the two plates be of an uniform thickness, and let the thickness of DF bear such a proportion to that of AB that the quantity of matter shall be the same in both. Consequently the quantity of matter in each part of DF will be very nearly equal to that in the corresponding part of AB. Also let the size of the plates be such that CE shall be very small in respect of the distance of C from the nearest part of the circumference of AB, and let the least radius of curvature of the surface of AB be so great in respect of CE that a point R may be taken such that CR shall be small in respect of that radius of curvature, and yet very great in respect of CE.

Let Pp be a flat circular plate whose center is G and whose plane is perpendicular to GZ, and let its area be equal to that of AB, and let the quantity of matter in it be also equal to that in AB, and let it be

[* Note 14.]

disposed uniformly the sum of the repulsions of AB and DF on CE in the opposite directions CE and EC will be to the repulsion of Pp on the infinite column GZ very nearly as 2CE to GP.

For suppose each particle of matter in all that part of AB whose distance from C is not greater than CR and in the corresponding part of DF to be transferred to its corresponding point in Tt, so as to form a circular plate whose radius is CR.

If we suppose that the thickness of the plates Tt and Pp are both equal to that of AB, the matter in all parts of Tt will be very nearly twice as dense as that in AB or as that in Pp. Therefore the repulsion of Tt on CE will be very nearly twice the repulsion of Pp on Gg, supposing Gg to be equal to CE.

But from the foregoing lemma it appears that the sum of the repulsions which the above-mentioned part of AB and DF exerted on CE before the matter was transferred is very nearly equal to that which 7t exerts thereon after the matter is transferred, and the sum of the repulsions of the remaining part of AB and DF, or that whose distance from C is greater than CR, is very small in respect of that part whose distance is less, therefore the sum of the repulsions of the whole plates AB and DF on CE is to the repulsion of Pp on GZ very nearly as 2CE to GP.

It may perhaps be supposed from this demonstration that it would be necessary that CE should be excessively small in respect of CV, in order that the sum of the repulsions of the plates on CE should be very nearly equal to the repulsion of Pp on Gg, but in reality this seems not to be the case, for if the plates are segments of concentric spheres whose center is V, the sum of their repulsions will exceed twice the repulsion CE of Pp on Gg in a not much greater ratio than that of 1+ to 1, and CV

if the radius of curvature of their surfaces is in some places greater than CV, and nowhere less, I should think that the sum of their repulsion could hardly exceed twice the repulsion of Pp in so great a ratio as that.

157] COR. II. If we now suppose that the matter of the plate AB is denser near the circumference than near the point C, and that the density at and near C is to the mean density (or the density which it would everywhere be of if the matter was spread uniformly) as d to one, and that the quantity of matter in each part of DF is equal to that in the corresponding part of AB as before, the sum of the repulsions of the plates on CE will be less than if the matter was spread uniformly in a ratio approaching much nearer to that of 8 to one than to that of equality.

For if any particle of matter is removed from that part of AB which is near to that point which is at a distance from it, and an equal alteration is made in the plate DF, the sum of the repulsions of these particles will be much less after their removal than before.

158] LEMMA XVII. Fig. 7. Let ACB be a thin plate, not flat but concave on one side, let the radius of curvature of its surface be

Fig. 7.

M

R

T

B

nowhere less than CV, and let MV be perpendicular to its surface at C'; let MC be very small in respect of CV, and let T't be a plane perpendicular to MC: the difference of the repulsion of any particle of matter as R in the plate ACB on the point M in the direction CM, and of its repulsion on the point C in the same direction, is very nearly the same as if the particle was transferred to T (CT being equal to the right line CR), provided CR is small in respect of CV.

MN
MR3 CR3

Draw RN perpendicular to MC, the difference of the repulsions of CN MC CN CN R on the points M and C = and the ᏟᎡ MR MR3 CR3 difference of the repulsions of the same particle placed at 7' on the same

=

and therefore is very small in respect of CR2 or MT2.

1 Therefore MR2 differs very little from MT, and from MR3

1

MT3

This being premised there are two cases to be considered.
First, if CR is considerably greater than MC, as

CR2 = MR' – MC2 – 2MC × C'N = MR2 ×

MC (MC+2CN))

MR2