GRAVITATION. 



fore C = — a'. Tlie fcgmcnt tlicrcfore intcrccjned be- 

 tween B and C, the line A C being ^, is — a' 



«4 xj H x\ 



3 



This alfo gives -— a', foi- the content of the whole fulid, 

 15 



■when .V = o, the fame value that was found by another me- 

 thod at VI. 



Now, if we fuppofe .y to be = A A', and to be given = b, 



the folid content of tlie fesfment becomes — a' — j ^ a-, b) 



15 



+ - f which mud bo made equal to the given folidity, which 



we (hall fuppofe m^, and from thiti equation a, which is yet 

 unknown, is to be determined. If then for ti. we put ti, we 

 have TT (r\ m' — ^ l>]ii' — ', b ) — m\ or ^S "' — i ^.«' — 



- -^i',andi/'-2i;K=.-^- \\b-. 

 ir • 4- 



The fimpleft way of folving this equation would be by the 

 rule of falfe pofition. In fome particular cafes it may be 



refolvedmore eafdy ; thus, if -^ \\^' = O' "^ — i ^' "' 



= o, and II- = I b; that is a] = s. b], or a — b x {%)l = ^' 



IX. — 1. If it be yeqii'ired to Jitid the equat'ton to the fuper- 



Jides of the folid of greateft atirad'ion, and alfo to the fedions 



of its parallel to any plane paffmg through the axis ; this 



can readily be done by help of ivhat has been dem-nflraled 



above. 



Let AHB {fig. 138), be a feftion of the folid, by a plane 

 paffing through A B its axis. Let G be any point in the 

 fuperilcies of the folid, G F a perpendicular from G on the 

 plane AHB, and F E a perpendicular from F on the axis. 

 Let A E = X, E F == c, F G = 'ii, then .v, z, and v are the 

 three co-ordinates by which the fuperficies is to be defined. 

 Let A B =:^ rt, E H = J', then from the nature of the curve 

 AHB, y'' =■ a\ x\ — i.r'. But, becaufe the plane G E H is at 

 right angles to A B, G and H are in the circumference of 

 a circle of which E is the centre ; fo that G E = E H = _)■. 

 Therefore E F^ 4- F G^ = E HS that is ='- + v" = y\ and 

 by fubftitution for ji^ in the former equation, z^ -t- tj" = 

 ciA'} — .v', or {x^ -f z' 4- "u") = fl' jr' which is the equa- 

 tion to the fuperficies of the folid of greateil attraftion. 



2. If we fuppofe E F, that is z, to be given =: b, and the 

 folid to be cut by a plane through F G and C D (C D be- 

 ing parallel to A B) making on the furface of the folid the 

 feftion D G C ; and if A K be drawn at right angles to 

 A B, meeting D C in K, then we have, by writing b for z 

 in either of the preceding equations, b' -f- -u' = a\xi — .v', 

 and 11" = a\ .r? — .v'' — b' for the equation to the curve 

 D G C, the co-ordinates being G F and F K, becaufe F K 

 is equal to A E or .r. 



This equation alfo belongs to a curve of equal attraftion; 

 the plane in which that carve is being parallel to A B, the 

 line in which the attraction is eltimated, and dillant from it 

 by tlie fpace b. 



Inftead of reckoning the abfcifla from K, it may be made 

 to be^in at C. If A E or C K = h, then the value of h is 



detemnined from the eqtration i' ;= j' oj — //, and if .y «= 

 b + a, II being put for C F, i'" = a< {h -j- v)\ — (h 4- «)" — 

 a)h. 4- fy, or v' + {h + u) " + b" — «} (A 4- u)', or {v"- -{- • 

 h + u)'+ If) '' = a" {h 4- uy. 



When b is equal to the maximum value of the ordinate 

 E H (IV. 2) the curve C D G goes away into a point ; and 

 if b be fuppofed greater than this, the equation to the curve 

 is impoflible. 



X. — The folid of greattjl allrai^ion may be found, and its 

 properties inve/ligatcd, in the ituiy that has tio'zv been cxem- 

 plijitd, iL'hatever be the hviu of the atiradi'oe force. It ivill 

 be fifficitnt in any cafe to find the equation of the generating 

 curve, or the curve of equal altra8:on. 



Tluis, if the attraftion, which tlie particle C [Jig. 136.) 

 exerts on the given particle at A, be inverlely as the m 



power of the diftance, or as-;^^-;;-, then the attraftioiv 



in the direftion A E will b( 



AC 



AE 



this = 



we hare • 



■ AC"-' 

 AE 



A B "■ A C " + ■ A E ■ 



A E = .V, E C = 3', and AB :=: a, as before 



=— , or <2"'.ir = (•» 4- /')"-V"' *"'!■' 



and if we maka 



or making 



(•'^+^^)- 



or y' =r -— .«-. 



If m ^= I, or )« 4- I = 2, this equation becomes y' =r a .v 

 — .v', being that of a circle of wliich the diameter is A B. 

 If, therefore, the attrafting force were inverfely as the dif- 

 tance, the folid of greateft attraftion would be a fphere. 



If the force be inverfely as the cube of the dillance, or 

 ni =: 3, and m 4- i = 4, the equation is y'' = a; xi — ,v' 

 which belongs to a line of the fourth order. 



If m = 4, and m + I = 5, the equation is y'' = a; .v] — .v', 

 which belongs to a line of the tenth order. 



In general, if m be an even number, the order of the curve 

 is n; 4- I X 2 ; but if m be an odd number, it is m 4- I fim- 

 ply. 



In the fame manner that the folid of greateft attraftion 

 has been found, may a great clafs of fimilar problems be 

 refolved. Whenever the property that is to exift in the 

 greateft or leaft degree belongs to all the points of a plane 

 figure, or to all the points of a folid, given in magnitude, 

 the queftion is reduced to the determination of the locus of 

 a certain equation, as in the preceding example. 



Let it, for inilancc, be required to find a folid given in 

 magnitude, fuch, that from all the points in it, ftraight lines 

 being drawn to any number of given points, the fum of the 

 fquares of the lines fo drawn ftiall be a minimum. It will be 

 found, by reafoning, as in the cafe of the folid of greateft 

 attraftion, that the fuperficies bounding the required folid 

 muft be fuch, that the fum of the fquares of the lines drawn 

 from any point in it, to all tJie given points, muft be always 

 of the fame magnitude. Now, tlie fum of the fquares of 

 the lines drawn from any point, to all the given points, may 

 be fhewn by plane geometry, to be equal to the fquare of the 

 line drawn to the centre of gravity of thefe given points, 

 multiplied by the number of po'iits togetlier with a given 

 fpace. The line, therefore, drawn from any point in the re- 

 quired fuperficies, to the centre of gravity of the given points, 

 is given in magnitude, and therefore the inperficics is that of a 



fphcFC, 



