ROTATION. 



If the body B be a fphere, whofe radius is r, then G Q 1 



= \r , and if a be the abfolute velocity of rotation of an 



u 

 equator of the fphere, we have U = ; whence the pre- 



ceding value of G S is transformed to this, G S 



a 



r -v 



This proportion may be applied to the double motion of 

 the planets. The earth revolves about an axis palling 

 through its centre of gravity, while, by a motion of 

 translation, that centre is carried on in free lpace in an orbit 

 nearly circular ; and a fimilar kind of double motion has 

 been difcovered in feveral of the planets, and analogy leads 



third axis lying in the plane of the other two, and inclined 

 to each of the former axes in angles whofe fines are inverfelv 

 as the angular velocities round them ; and the angular ve- 

 locity V round this new axis, is to that about one of the 

 primitive axes, as the fine of the inclination of the latter 

 axis, to the fine of the inclination of the new axis to the 

 other primitive axis. 



Thus.ifabodyturns round an axis AGa,(Jg. 10.) palling 

 through its centre of gravity G, with the angular velocity U. 

 while this axis is carried round another axis BGi, with the 

 angular velocity a ; and if G D be taken to G E as U to u, 

 (the points B and E being taken on that fide of the centre' 

 where they are moving towards the fame fide of the feu 



oeen ancoverea in levenu 01 uie piaucis, aim analogy leaus vvucit: tury are moving towards tne lamellae or the fieure r 

 us to believe it obtains in the others. Now, fuppoling the and the line D E be drawn, the whole and every particle of 

 bodies of the planets to be fpherical, as they are nearly, the the body will be in a (late of rotation, about a third axis 



ufe of this propofition at once appears. Having given, for 

 inftance, the magnitude of an impulfe, with refpeft to the 

 mafs of the earth, and the direction ; S, in which it was ap- 

 plied, at any given diftance S G from the centre of gravity, 

 the angular motion round G would be inferred ; and con 



C G c parallel to D E, lying in the plane of the other two, 

 and the angular velocity -v about the axis C Gr, will be to 

 U as D E : D G ; and to u as D E : G E. For let P be 

 any particle of the body, and fuppofe a fpherical furface, 



. whofe centre is G, to pafs through P. Draw P R perpen- 

 , dicular to the plane of the figure ; then is P R the common 





verfely if the adual rotatory velocity of the earth's equator, uicuiar to tne piane or tne liguic 



and the velocity in its orbit, be afcertained, the diftance feftion of the circle of rotation I P ; round the axis A a and 

 G S from the centre, at which it may have received a (ingle the circle K Pi of rotation round the axis lit. Let F and O 

 impulfe ; S adequate to produce the double motion, may be be the centres of thefe circles of rotation, and I ;', and K t 

 readily found. their diameters. Draw the radii P F, P O, and the tan- 



Thus, any point in the earth's equator pafies over 25,020 gents P M, P N ; thefe tangents are in the plane M P N 

 miles by rotation in one iidercal day ; and if the mean dif- which touches the fphere in P, and the plane of the axis in 

 tance of the earth from the fun be 95 million miles, the the line M N, to which a line drawn from G, through R 

 earth will pafs over nearly 596,904,000 miles by its orbital would be perpendicular. Suppofe P N to reprefent the ve- 

 locity of rotation of the point P, about the axis B&, while 

 P/reprefents its velocity of rotation about A a, and com- 

 plete the parallelogram P N //; then is P / the diredion and 

 velocity of the refultant of the compofition of PN, P/, and 

 it is manifestly in the fame plane as the conftituent lines.' Let 

 perpendiculars/F, /T, be drawn to the plane of the axis, 

 and the parallelogram PN;/ will be orthographically pro- 

 jected on that plane, its projeaion being alfo a parallelogram 

 R N T F. Draw the diagonal R T. Then, iince PR ii 

 perpendicular to the plane of the primitive axis, P R / T is 

 fo likewife; and confequently the compound motion Pr is 

 in the plane of a circle of revolution about fome axis lituated 

 in the plane of the other two. Produce I R, and draw 

 G C interfecting it perpendicularly in H ; and let L P/ be 

 the circle of rotation, its diameter being L / — 2 L H ; then 

 is P/ a tangent, and perpendicular to P H, and it will meet 

 IR in fome point Q of the line M N. The particle P is in 

 a (late of rotation about the axis C Gr; its velocity is to tho 

 velocities round A a, or B t, as P / to P F, or P T to P N. 

 Now PN the tangent is perpendicular to O P, and P R ip 

 perpendicular to O N ; therefore OP:PN::PR:RN, or 



PN. R N 

 — ^^. But the velocity of P about the axi» 



motion in a year, or iH about 366 (idereal days ; hence 

 596,904,000 -£■ 366 = 16,308,852, will be equal to V 

 in our theorem, while 25,020 = u. Confequently, G S 



, . So that if an impulfe be imprefTed 



163 .2 



upon a quiefcent fphere, and the direction of force fliould 



be at a diftance S G from its centre of gravity of about -,-j-rd 



part of its radius, the angular motion of that fphere, and 



the abfolute motion of its centre, will have the fame relation 



to each other, as thofe which actually obtain in the earth. 



The time of the rotations of Mercury, Uranus, and 



the laft new planets, are unknown ; but for the following 



planets it is aicertained, fo that, by the fame theorem, we 



; Jupiter, 



195 * 



obtain thefe values of G S ; viz. Mars, 



r r >' 



— r ; Saturn, — — - ; the Moon, ■ — . . 



2.8125 2.588 555 



We have not fufficient data for the fun ; but the very 

 circumflance of his having a rotation of 27 11 7 U 47™, makes 

 it very probable that he, with all his attendant planets, is 

 alfo moving forward in celeftial fpace, perhaps round fome 

 centre of dill more general and extenfive gravitation ; for 

 the perfeft oppofition and equality of two forces, for giving 

 a rotation without a progreffive motion, has the odds againll 

 it of infinity to unity. This corroborates the conjectures 

 of philofophers, and the obfervations of Herfchel and other 

 aftronomers, who think that the folar fyflemis approaching 

 to that quarter of the heavens in which the conftcllation 

 Aquilea is fituated. 



Prop. IV. 



If a body revolves about an axis palling through its centre 

 of gravity with the angular velocity U, while this axis is 

 earned round another axis, alfo paffing through its centre 

 of gravity, with the angular velocity U, thefe two motions 

 compofe a motion of every particle of the body, round a 



Vol. XXX. 



PO 

 B b, is u . O P ; whence R N = ^-L^Z.!^ = u . P R. 



In like manner, R F = U . P R ; confequently R F : R N 

 :: U : u :: G D : G E. But N T : R N :: fin. NRT: fin. 

 N T R :: fin. G E D : fin. G D E ; hence fin. NRT: fin. 

 N T R :: fin. G E D : fin. G D E. Now, fince N R is per- 

 pendicular to E G, and NT (parallel to F I) perpendicular 

 to D G, we have R N T .= E G D. Hence T R is perpen- 

 dicular, and Cr parallel to ED., Alio, lime R N, R F, 

 R T, are as the velocities u, U, v, about thefe different axes, 

 and vary refpeftively, as E G, D G, D E, we have r : U : 

 u::ED:GD:GEj which w.i!- n. be demonftrated, 



Hence, if every particle of a body, whether folid or 



fluid, receives at the fame time two feparate impulfcs, the 



4 1 one 



