G R A 



G R A 



fpkeTc, having for Tts centre the centre of gravity of the 

 given points. 



The magnitude of the fphere is next determined from the 

 condition that its folidity is given. 



In general, if .\;y and z are three reftangular co-ordinates, 

 that detern'.ine the pofition cf any point of a foUd given in 

 magnitude, and if the vahie of a certain function 2 of .r, jp, and 

 2 be computed for each point of the foHd, and if the fum 

 of all thele va!;:es of * addjd together be a maximum or a 

 minimum, the folid is bounded by a fuperlicies, in which the 

 function ; is every where of the fame magnitude. Th?t is, 

 if the triple integral /.v_/' / ? i. be the greatoll or leafl; pof- 

 fible, the luperficits boundmg the folid is fuch, that f =. A, 

 a conil;.nt qizaniity. 



The fame holds of plane figures. The propofition is then 

 more finiplc, as there are only two co-ordinates, fo xXvMfx/z k 

 is the quantity that is to be a maximum or mmimum, and 

 the line bounding the figure is defined by the equation 

 <5 = A. 



AM the queilions therefore that come under this defcrip- 

 tion, though they belong to an order of problem", which 

 requires in general the application of one of the moll refined 

 inventions of new geometry, the calculus •uariationum, form 

 a particular divifion, admittipg of folution by much more 

 fimple means, and direclly reducible to the conftruction of 

 loci. 



In thefe problems alfo, the fyntheticil demonftration will 

 be found extremely iimple. In the inftance of the folid of 

 greateft attraction, this holds remarkably. Thus it is ob- 

 vious, that (f^g. 136.) any particleof matter placed within the 

 curve A C B iri, will attrod the particle at A in the direc- 

 tion A B, lets than any of the particles in that curve, 

 »nd that any particle of matter witliin tlie curve will at- 

 tract the particle at A more than any particle in the curve, 

 and more, a fart'wrl, than any particle without the curve. 

 The fame is true of the whole fupL-rhcies of the folid. Now 

 if the figure of the fohd be any how changed while its 

 quantity of matter remains the fame, as much matter muft 

 be expelled from within the furface, at fome one place C, as 

 is accumulated without the furtace, at fome other point H. 

 But the aftion of any quantity of matter within the fuper- 

 ficies A C B H on A is greater than th.e aftion of the fame 

 without the fuperficies ACBH. The folid ACBH 

 therefore by any change of its figure mult lofe more attrac- 

 tion than it gains. Thus is its attraction not diilinguiflied 

 by every fuch change, and therefore it is itfelf the folid 

 of grealeli attraction. 



Among a number of propofitions which the limits of our 

 work do not permit us to notice, it is propofed, " to deter- 

 mine the oblate fpheroid of a given fohdity, v.hich fliall at- 

 tract a particle at its pole with the greatelt force." And it 

 appears that the gravitation at the pole of an oblate fpheroid 

 is not a maximum, until the excentricity of the generating 

 eliipfe vanilh, and the fpheroid pafs into a fphere. When a 

 fphere paiTes into an oblate fpheroid its attraction varies at 

 firll exceeding (lowly, and continues to do fo till its cblate- 

 nefs, or excentricity, becomes very great 



The cone of greateft attraction has the radius of its bafe 

 aearly double that of its altitude, and the attraction of the 

 cone, when a maximum, is about f the attraftion of a fphere 

 of equal folidity. 



Of all the cylinders given in mafs or quantity of mat- 

 ter, that which attracts a particle at the extren-.ity of its axis 

 with the greateft force is when the radius of the bafe of the 

 cylinder is to the altitude as five to eight nearly, and it ap- 

 pears that the attraaion of the cylinder, even when its form 

 is the moft advantageous, does not exceed that of a fphere of 



V»t. XVI. 



the fame folid cootent, by nwre than a hundred and cighljr- 

 third part. 



A femi-cylinder given in magnitude, attracts a particle 

 fituatcd in the centre of its bafe, with the grcatell force 

 poffible, in the direction of a line bifcdting the bafe, »he« 

 tlse altitude of the lemi-cylinder is to the radius of its bafe 

 as 125 to 216. 



Gr.witation, Centre rf. SeeCESTEli. 

 Gravitatio.s', Line nf. See LiNE 

 Gravitation, Plane of. Sec Pl.we. 

 GRAVITY, Terke-trial, is that force by which all 

 bodies are continually urged towards the centre of the earth. 

 It is in confequcnce of this force that a body cannot remain 

 at rell on the furface of the earth, without cxercifing a pref- 

 fure either on fome intermediate body, or on thit {onion of 

 the furface of the earth which fuftains it ; and the inttr.fity of 

 this force is mcafurcd by the degree of prclTure, produced by a 

 given mafs. The tenfion of a llring, by which a weight is fuf- 

 pended, arifes from the force of gravity. The fpring fteel- 

 yard, an inftrument fold at the ihops for weigh;rg, is ex- 

 tremely well adapted to iUuftrate the cSect of the force of 

 gravity ; the fufpended fubltance draws out a fcale b/ over- 

 coming the refiftance of the fpring. Thefe machini* -ire not 

 capable of great exadtnefs, but if an inltrumcr.t of this kind 

 could be made with fuiSciert accuracy, the alteration of the 

 fcale, (the weight remaining the fame,) would flicw any 

 change in the force of gravity ; and we might, by taking 

 this apparatus to the fumn-.it of a high mountain, ob- 

 ferve whether any change took place in the force of gra- 

 vity bv fuch an operation. We meafure like«ife the force 

 of gravity by the time which a bod), fuffered to defcend 

 freely from a ftate of reft, employs to fall thiough a given 

 fpace, or (as has been explained under DvSA.MlCs) by the 

 velocity which a body, thus faUing, acquires at the end cf a 

 given time. Thus a body, filhng freely, durinij the interval 

 of one fecond, acquires a velocity of 32^, that is, it would 

 ftiike an obftacle with the fame force as another body 

 would of the tame mafs, which was monng uniforndy with 

 the velocity of 32 s per fecond. 



The denVity of the earth being about 2.5 that of water, 

 we infer that the force of gravity of a fphere, 80CO mile* 

 in diameter, and whofe fpecihc gravity is twice and a half 

 that of water, would attract a particle of matter placed jull 

 without its furface, in fuch a manner as to caufe it to move 

 towards its centre, 16,-., feet in one iecoiid. 



It is, however, to be obferved, that it is only at the pole 

 that the whole force of the earth's gravity is aftually ex- 

 erted on a particle of matter : at every other part of the 

 ear h's furface, the force of gravity is dimiiiilhed by the 

 motion of rotation, which producing a centrifugal force, 

 oppofite in its tendency to that of the force ul gravity, 

 diminifhes the effect of the latter as we recede from the^)oIe 

 and .ipproach the equator. But, befides this, there is an- 

 other caufe which contributes in a very remarkable maimer 

 to modify the force of gravity at dilTerent points of the 

 earths tarface, which is the elliptic figure of the eartlu 

 The equatorial regions being more ele\-ated than the polar, 

 are more removed from the influence of tlie central attrac- 

 tion. Both thefe cirrjmftances, and their cffcAs, muft be 

 attended to when we propofe to m.ake any very accuiate 

 computation of the force of terreftiial gra\ity. 



The diminution of gravity arifing from ihe elliptic 6gire 

 is nearly equal to the product of the .^.jtli part ot the Jorce 

 of gravity, bv the fquare of the cofiue of the latitude. 

 The centrifugal force diminilhcs the force of gravity m the 

 fame proportion ; thus by the combination of thefe two 

 caufes, the diminution of gravity from tlse pole to die 

 4 R equaisr 



