G R A V IT A T I O N. 



Hence -— , or the radius of curvature at B = ? a. The 



2 3 



curve therefore at B falls wholly without tlie circle B K A, 

 defcribed on the diameter A B, as ita radius of curvature is 

 greater. This is alfo evident from the conftruclion. 



V. To find the force 'wllh -which the fol'ul above defined attracls 

 the particle A in the dirctlion A B . 



Let B {jig. isyO) be a point indefinitely near to B, and let 

 the curve A ci be defcribed fimilar to A C B ; through C 

 draw C f D perpendicular to A B, and fuppoie the figure 

 thus conilruAed to revolve about A B ; then each of the 

 curves A C B, Aci, will generate a folid of greatcft at- 

 traftion; and the excefs of the one of thefe folids above the 

 other, will bean indefinitely thin (hell, the attraftion of which 

 is the variation of the atlraiflian of the folid A C B when 

 it changes into Act. 



Again, bythehneDC, when it revolves along with the 

 reft of the figure about A B, a circle will be defcribed ; 

 and by the part C c, a circular ring, on which, if we fuppofe 

 a fohd of indefinitely Iniall altitude to be conftituted, it 

 will make the element of the fohd fliell AC c. Now the 

 attraction exerted by this circular ring upon A will be the 

 fame as if all the matter of it were united in the point C, 

 and the fame, therefore, as if it were all united in B. 



But the circular ring generated by C f is = n- (D C^ — 

 D c') = 2 V D C X C <:. Now, 2 1) C X C c is the va- 

 riation of jji% or D C", while D C paffes into D c, and the 

 curve B C A into the curve 1/ c A; that is, 2 D C x Cc is 

 the fluxion of y', or of a' x] — x', taken on the fuppofition 

 that X is conftant and;? variable, namely i a[ a x .v;. There- 



fore the fpace generated hy C c ^ ^^^— a] x' d. 



3 

 If this expreffion be multiplied by x, we have the ele- 

 ment of the (hell = — - a x-] d x. 

 3 

 In order to have the folidity of the fliell AC B i f , the 

 above expreffion mull be integrated relatively to x, that is, 



4- „ 



fuppofing only x variable, ?nd it is then ^ x 



+ c. 



But C = o, becaufe the fluent vaniflies when .r vaniflies, 

 therefore the portion of the ftiell A C c ^= | *•; a- a, and 



when .V =: a the whole (hell = - — a^ d. 



Now, if the whole quantity of matter in the fliell were 

 united at B, its attraftive force exerted on A would be the 

 fame with that of the ihell, tiierefore the whole force of 



the fliell = — a. The fame is true for every other inde- 



5 

 finitely thin fliell, into which the iolid may be fuppofed to 

 be divided ; and, therefore, the whole attraction of the folid 



is equal to I— — a, fuppofing ^ variable, that is, = — a. 



Hence we may compare the attraftion of this folid with 

 that of a fphere of which the axis is A B, for the attraction 



of that fphere = -^a' x ~= —^ X a : the attra£lion 



6 a' 3 



of the folid A D B H (fg. 136), is, therefore, to that of the 



fphere on the fame axis as i-^ a 

 5 



to a, or as 6 to 5. 



3 



VI. Tofnd the content 0/ the folid A D B H, we need only- 

 integrate the fliixionary expreffion for the content of the 



fliell, namely, —■ d^ a. We have then 



test of the folid A D B H, Since 



— a = the con- 

 the folidity of the 



fi'hcre on the axis A is = — a\ l)ie content of the folid 

 o 



A D B H is to that of the fphere on the fame axis as— a' 



to — a^ ; that \s, as — to /, or as 8 to c. 



VII. Laflly, to compare the attraaion of thii firul lutlh the 

 aftraaion of a fphire of equal bulk. — Let m — any given 

 mafs of matter formed into the folid A D B H ; then for de- 

 termining A B, we have this equation -— a' = m\ and a = 



.''5 



>5 



"'' ^ — ; and, therefore, alfo, the attraction of the foHJ, 



4~ 



'5 



4- V 4-5 / 



Again, if m' be formed into a fphere, the radius of that 

 fphere = m' '' — , and the attradlion of it on a particle at 



Its lurlace = m- 



Hence tlie attraflion of the folid A D B H, is to that of 

 a fphere equal to it, as m ( i- t" V to m t — x' V ; that 

 is, as (27)', to (25)', or as 3 to the cube root of 25. 



The ratio of 3 to ^25, is nearly that of 3 to 3 ^ 



or of 8 1 to 79 ; and this is therefore alfo nearly equal to the 

 ratio of the attraction of the fohd A D B H, to that of a 

 fphere of equal magnitude. 



VIII. — // has been fuppofed in the preceding invefligation, 

 that the particle on 'which the folid of grealejl attraaion exert t 

 its force, is in contaa with that folid. Let it no-w be fuppofed-, 

 that the diflance betiveen the folid and the particle is given ; the 

 foHd being on one fid. of the plane and toe p<ir:icle al a gi^ft 

 diflance Jrom the fame plane on the oppofite fide. The mafs of 

 matter -which is to compofe the folid being given, is is required /» 

 confirud the folid. 



Let the particle to be attracted be at A fjig. r37 ), from .\ 

 draw A A' perpendicular to the given plane, and let E F be 

 any ftraigiit line in that plane drawn through the point A', it 

 is evident that the axis of tlie folid required mufl be in A A' 

 produced. Let B be the vertex of the folid, then it will 

 be demon (Irated as has been done above, tliat this foliJ is ge- 

 nerated by the revolution of the curve of equal attradion. that 

 of wliich the equation is {y := a\ x — x) about the axis of 

 which one extremity is at A, and of which the length muft 

 be found from the quantity of matter in the folid. 



The folid required then, is a fegment of the folid of grcat- 

 efl; attraftioii, having B for its vertex, and a circle of which. 

 A' E or A' F is the radius, for its bafe. 



To find the folid content of fuch a fegment, C D be- 

 ing = y, and A C = .«, we have y = a x- — .r, and 

 T. y' x — ■rr a' X} x — z- .v' x = the Cylinder, which is the ele- 

 ment of the folid fegment. 



Therefore/:?/ .V, or the fohd fegment intercepted l>e- 

 tween B and D, muft be i :t a! x] — i ■* .r + C. This 

 muft vauifli wheu .v = <j, or when C comes to B, and there- 



tux« 



