ROTATION. 



fufpenfion, the correfponding centre of ofcillation will be 

 the centre of fpontaneous rotation. This follows imme- 

 diately from the properties of the centre of percuffion and 

 ofcillation. (See Center.) To which we may alfo add, 

 that fince the force divided by the body afted upon, is a 

 general expreffion for the velocity of the centre of gravity 

 of that body ; therefore the velocity of the centre of 

 gravity of the body will be the fame, whatever may be the 

 direction of the impelling force ; fo that the permanency of 

 the quantity of motion obtains the fame in motions about 

 a centre of fpontaneous rotation, as in all other cafes. 



Prop. I. 



When $S, (jrg- 9-) the direction of impact, pafTes 

 through the centre of the impelling body, the centre 

 of gravity of the body ftruck, will move with a velo- 

 city equal to the product of the quantity of motion of 

 the impelling body into the diitance between the centre 

 of gravity and fpontaneous rotation, divided by the fum 

 of the products of the impelling body into the diftance 

 of the point of impact from the centre of fpontaneous rota- 

 tion, and of the impelled body into the diftance of the im- 

 pelled body, into the diftance between the centres of fpon- 

 taneous rotation and of gravity. 



Let the quantity of matter of the impinging body be b, 

 its velocity v, or bv — <p; and when the body B is ftruck 

 in the direction <a S, (in which the centre of the body b 

 is always found), let the velocity of its centre of gravity 

 be V, the centre of fpontaneous converfion being at C. 

 Then C G : C S :: V : the velocity of the point S, which is 



therefore V . ; confequently, v — V . n r ,~ 



the ve- 



locity loft by b in the direction $ S ; whence, by the third 



. c . t,.GC-V.CS _ _ ,, 

 law or motion, * x 7^r^r~ = r> . V, and by re- 



duction, V 



CG 

 CG 



as was to be demon- 



B.CG+i.CS' 

 ftrated. 



Hence, if the inertia of the ftriking body be evanefceut, 



the velocity V will become = -=-, being the fame as 



would be generated if the body b impinged directly on it 

 with the velocity V. 



Prop. II. 



The inertia of the ftriking body being evanefcent, the 

 angular velocity of the fyftem about the centre of gravity 

 is equal to the momentum of the impelling body, divided 

 by twice the product of the mafs of the impelled body, and 

 the diftance C G, into the periphery of a circle whofe 

 diameter is unity. 



If the fixed axis paffed through C, the centre of gravity 

 would defcribe a circle, whofe radius is C G, with the 



b v 

 velocity --. But, in the prefent cafe, the motion of the 

 B 



fyftem will be compounded of the uniform reftilinear motion 

 of the centre of gravity, in the direction G C, perpen- 

 dicular to C S, and the angular motion x G'p = 6 C G', 

 generated round the centre of gravity. And fince the 

 periphery of a circle, whofe radius is C G, is 2t.CG 



. b v 



(r being = 3.1416), we have this analogy, -^ : 2 it . C G 



iT.E.CG 1 

 b v 



, the time of one revolution in feconds. 



Whence it follows, that the number of revolutions, or parts 

 of a revolution, in a fecond, or the angular velocity U, 



.... 2 v . B . C G bv 



will be 1 -; = . 



bv 2 - . d . C G 



And fince C is the centre of percuffion to S as a centre 

 of motion, if Q be the centre of gyration with refpeCt to G 

 as a centre of motion (that is, if Q be the principal centre 

 of gyration), we ihall have GC.GS= G-Q-, or C G = 



G O" 



7^-j,-. This value of GC being fubfhtuted for it in the 



preceding expreffion for the angular velocity, it becomes 



U =: — — ■ . It follows alfo from what is (hewn above, 



2 it B . G Q~ 



that the centre of fpontaneous rotation, during the motion 

 ef the fyftem, defcribes the common cysloid. For the 

 motion of any point in the fyftem is compounded of the 

 uniform rectilinear motion of the centre of gravity, and of 

 the angular motion generated round that centre ; but the 

 velocity with which the centre of fpontaneous rotation 

 would move round the centre of gravity, if there only 

 exifted a rotatory motion in the fyftem, would be equal to 

 that with which the centre of gravity would move round 

 it, if the centre C were fixed ; confequently, fince the 

 centre C has both a rotatory and progrellive motion, each 

 of which is equal to that of the centre of gravity, it will 

 defcribe a cycloid. 



Prop. III. 



In the body, or fyftem B, {Jig- 9.) to which, when 

 quiefcent, motion has been communicated by the impulfe 

 of a force, <p, without inertia, that is, rectilinear motion 

 to the centre of gravity meafured by the fpace V, which 

 that centre would defcribe uniformly in any given time, 

 and angular motions meafured by the revolutions U, or 

 parts of a revolution, which it would defcribe uniformly 

 round G in the fame time ; then if the notations in the pre- 

 ceding propofitions be retained, and Q be the principal 

 centre of gyration, when the fyftem revolves about its 

 centre of gravity, the perpendicular diftance from the cen- 

 tre of gravity, at which the impelling force rauft act fo as 

 to have generated thefe progrellive and rotatory motions, 



wiUbeGS = i^^: 



Let <p S be the direction of the impulfe, and let $ be 

 equal to the momentum of an evaneloent body b, moving 

 with the velocity V, B being the weight of the fyftem ; 

 then the velocity communicated to the centre of gravity of 



20 /"^ 



the fyftem, will, by the laft propofition, be = ' . 



2-.B.GQ- 



But by the propofition, the velocity communicated to the 

 centre of gravity of the fyftem is V ; and the angular mo- 

 tion, that is, the number of revolutions, or parts of a revo- 

 lution, defcribed while the centre of gravity pafTes over the 

 fpace V, is U ; fo that from the conditions there arifes this 

 b.v.GS V.GS 



equation, U = ^ 



2 7T.B.GQ/ 2*.GQ-' 



by putting Y 



b v 

 for its equal . 



Hence G S-.=c 



2t.U.GQ ! 



If 



