GRAVITATION. 



= A B- X A E ; nnd as tliia equation is fufficient to deter- 

 mine the nature of the cm-veto which it belongs, therefore 

 all the feftions of the folic!, by planes that pafs through A B, 

 are fnnilar and equal curves ; and the folid, in confequeoce, 

 may be conceived to be generated by the revolution of 

 A C B, any one of thofe curves, about A B as an axis. 



The folid fo generated may be called the foUd of gnatejl 

 attraaioiiy and 'the line A C B the cum of equal attrac- 

 tion. 



II. To find the equation bctivecn the co-ordinates of A C B, 

 the curve of equal attra^ion. 



From C ( fig- 136.) draw C E perpendicular to A B ; let 

 AB = <2, AE = *, £€=;-. 



It has been found that A B^ X A E = A C% that is 

 a' X = (x- + y')], or a* x' — {x' + /')', which is an equa- 

 tion to a hne of the fixth ord^r. 



To obtain y in terms of x, x' -\- y" — a] x], y = a\ x] — *'' 

 and y = X'- \'a'- — x\ 



Hence ji = o, both when « = o, and when r = a. Alfo, 

 if X be fnppofed greater than a, y is impoffible. No part 

 of the curve, therefore, lies beyond B. 



The parts of the curv'e on oppofite fides of the line A B 

 are fimiliir and equal, becaufe the pofitive and negative va- 

 lues of y are equal. There is alfo another part of the 

 curve on the fide of A, oppofite to B, fimilar and equal to 

 A C B ; for the values of y are the fame, whether x te 

 pofitive or negative. 



III. The curve may le eafdy conJJrudcd without having 

 recourfe to the value of y jujl obtained. 



Let A B = a [fg. 136.) AC = z, and the angle BAG 

 .= ^. Then A E — A C X cof. (p = 2 cof. if, and fo 

 a' c cof. !j> = z', or a" cof. 0=2"; hence z. ^^ a ^/ cof. (p. 



From this formula -a value of A C or z may be found, if 

 ^ or the angle B A C be given ; and if it be required to find 

 z in numbers, it may be conveniently calculated from this 

 expreOion. A geometrical conftruftion may alfo be eafily 

 derived from il. For if with the radiuE A B, a circle B F H 

 be defcribed from the centre A ; if A C be produced to 

 meet the circumference in F, and if F G be drawn at right 



A G 



angles to A B, then •^— - = cof, ip, and fo z = iz x 



a'- ?, and fnice A E"- : A C-_: : fcft. A E G : feft. ACT, 

 the fedor A C F = ■§ z' ^. But z' = a" cof. ^, whencs 

 the. feftor AC F=i z^ ?>. But z''=a' cof. ?i, (III.) whence 

 the feftor A C F, or the fluxion of the area A B C = i a" 5 

 cof. (?, and confequently the ai'ea A B C = 4 <t' I'u. y, 

 to which no coiiilant quantity need be added, becaufe it va- 

 nilTies when ((> = o, or when the area ABC vanillies. 



The whole area of thif curve therefore is \ a, or i A B' ; 



for when ^ is a riglit angle fin. ?i = 1. Hence the area of 

 the curve on both fides of A B is equal to the fquare of 

 AB. 



2. The value of.v, ivhen _)> i'a maximum, is eafily found. 

 For when J', and therefore _)>■, is a maximum, | a^ — .Vj^^n^z .v, 



, . a a 



or \ x\ := a-, that is, *• = — = — 



Hence calling h the value of y when a maximum, Ir = 



a-. a- , /27I- i\ 2a' 

 a\ X = a J = , and lo * = 



ay; 27r \ 27.; / V^7 



V ^5= ^''A B X A G = A C. 

 ■ ^ A B 



Therefore, if from the centre A, with the diftance A B, 

 a circle B F H be defcribed ; and if a circle be alfo defcrib- 

 ed on the diameter A B as A K B, then drawing any line 

 A F from A, meeting the circle B F H in F, and from F 

 letting fall F G the perpendicular on A B, interftfting the 

 femicircle A K B in K ; and if A K be joined, and A C 

 made equal to A K, the point C is in the curve. 



For A K = ^^A B X A G, from the nature of the fe" 



mi-circle, and therefore A C = '^ A B x A G, which has 

 been fhewn to be a pro^^erty of the curve. In this way a 

 number of points of the curve may be determined ; and the 

 folid of greatejl atlracliow will be defcribed, as already ex- 

 plained, by the revolution of this curve about the axis A B. 



IV. To find the area of the curve A C B. 



I. Let ACE.AFG (fg-i^T-) be two radii indefinitely 

 near to one anotiier, meeting the curve A C B in C and F, 

 and the circle defcribed with the radius A B in E and G. 

 Let A C = z as before, the angle B A C = .;, and 



A B = a. Then G E= a (?, and the area A G E = i 



61- 



/ 2 

 a , and therefore a : b : : i/ zi : ./2, or as 1 1 : 7 



*/27 



nearly. 



3. It is material to obferve, that the radius of curvature 



v^ a' 

 A is infinite : for fince v' =: a' xi — x',— = — ■ x. 



X XI 



But vihen X is very fmall, or y indefinitely near to A, — 



X 



becomes the diameter of the circle, having the fame curvature 

 with AC B at A ; and when x vaniflies, this value of— , or 



— '■ .^becomes infinite,becaufe of the divifor.t-t being in that 



cafe =^ o. The diameter, therefore, and the radius of cur- 

 vature at A, are infinite. In other words, no circle, having 

 its centre in A B produced, and paffing through A, can be 

 defcribed with fo great a radius, but tliat at the point A it 

 will be within the curve of equal attraction. 



The folid of greateft attraction then, at the extremity of 

 its axis, where the attracted particle is placed, is exceedingly 

 flat, approaching more nearly to a plane than the fuperficies 

 of any fphere can do, however great its radius. 



4. To find the radius of curvature at B, the other extre- 



tremity of the axis, fince y' = ai .vj — .v, if we divide by 



y- a^x]— X- 



a — .f, we have — :- =: — • 



a — .r a — x 



But at B, when a — x, or the abfcifla reckoned from B 



vaniflies, — — is the diameter of the circle, having the fame 

 a—x 



curvature with A C B in B. But when a — .1= o, or a 



= X both the numerator and denominator of the fradlion 



— ^ '— vaniflies, fo that its ultimate value docs not ap- 



a — X 



pear. To remove this diflficulty, let a — x =^ %, or x =. a 



— z, then we have ji- = (a — z)f — (a — z)-. 



But when z is extremely fmall, its powers higher than the 



firft may be rejefted ; and therefore (a — ^Y^z^a' ( i ) ^ 



=: a; [1 &c. \ Therefore the equation to tl>e curve be- 

 comes in this cafe, v'' = a' X aW 1 ~ J — a" + 2 a a 



=: a' — 3 a s; — a" -)- 2 a 2 = J a a. 



Hence 



