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 2CK 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 and the real quantity of redundant fluid CK nearly as to nat. log. CB 2CK 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 2CK CK* that of 2 to nat. log. and that of 1 to nat. log. CB CB 152] PROP. XXXII. Fig. 4. Fig. 4. Let ADFB and adfb be two equal cylinders whose axes are EC and ec, let them be parallel to each other G 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 съ EC a ratio between that of log to log + log 2EC CB 2EC CB log to log + log EC EC + Eb съ CB provided the redundant fluid in the 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 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 that of log to log + logGg 2 EG 2EG 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 EC + Eb* to log + log съ 2EC 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.] 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. 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, moreCE2 over, 2 × (TE – TC) is greater than TE Now the repulsion of the point R on the column CE in the direc1 + ᎡᎬ - ᎡᏟ tion CE is and the repulsion of the point S on RC RE' X 1 = + + RC RE SC × SE SC × SE RC RE SC SE 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 TE × TC' - 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]. small in respect of TE- TC, and Moreover, as EM and SN differ very little from each other, D is very the repulsion of T. RC-SE Moreover, is less than or than and RC ᎡᏟ CV' RE is hardly greater than RM-CN and is therefore still less than RC-SE ; each differ from one in a less ratio than that of SE SC Therefore the sum of the repulsions of R and $ 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*. 156] 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.] |