DYNAMICS. 



297 



PLATE 

 CCXLI. 

 Fig. 12. 



Application. The increments on the velocity, in describing these 

 ' small spaces, will be as the product of the accelerating 

 force by the time. Now, the velocities at A and B be- 

 ing equal, the times of describing AD and BE will be 

 as the lines AD and BE. Again, the centripetal forces 

 at A and B being equal, will be as the equal lines AD 

 and BF ; but the body at A is accelerated by the whole 

 of the centripetal force ; whereas at B, the centripetal 

 force is to that which accelerates the body along the 

 curve, as BF to BG ; hence, the accelerating forces at 

 A and B are as AD and BG. Hence, the increments 

 on the velocities, in describing these small spaces, will 

 be as AD', and BG x BE. But by similar triangles, 

 BG:BF::BF:BE; and therefore BGxBErrAD'. 

 Therefore, the increments are equal, and consequently 

 the velocities at D and E equal. Now, since, in ma- 

 king equal and infinitely small approaches toward the 

 centre, the increments of velocity are equal, it follows 

 that, in making equal finite approaches, the increments 

 will be equal, and therefore the velocities acquired will 

 be equal. 



PROP. XV. 



If the orbit which a body describes in consequence 

 of a centripetal force, and which we may call the sim- 

 ple orbit, revolve from whatever cause in its own plane 

 around the ceatre of attraction with an angular veloci- 

 ty, which bears a constant ratio to that of the body, the 

 compound orbit which the body describes in conse- 

 quence of this compound motion will be deflected to- 

 ward the same centre of attraction. 



For, in the compound orbit, the angular velocity of 

 the body will be every where equal to what it had in 

 the simple orbit, plus or minus the angular velocity of 

 the orbit, according as the orbit revolves in the same 

 direction as the body, or in a contrary one ; hence 

 (from the constant ratio mentioned in the enunciation) 

 it is clear, '.hat the angular velocity in the compound 

 orbit will be every where proportional to the angular 

 velocity in the simple orbit ; but the angular velocity 

 in the two orbits will evidently h# proportional to the 

 momentary increase of area, the distances being the 

 came. Hence, the areas in the two orbits will increase 

 at the same rate ; but in the simple orbit, the areas are 

 proportional to the times, (Prop. 1.) Hence this will 

 also hold in the compound orbit, and hence this orbit 

 may be considered as having the same centre of attrac- 

 tion, (Prop, t.) 



PROP. XVI. 



The same supposition being made as in last proposi- 

 tion, the difference at any point betwixt the centripetal 

 force that acts on the body in the simple orbit, and that 

 which acts on, or will be necessary to retain the body 

 in the compound orbit, will l>e inversely, as the cube 

 of the distance of the point from the centre of attrac- 

 tion. 



To simplify the expression of the reasoning, suppose 

 that the body is approaching nearer to the centre of at- 

 traction. It is evident, that the approach made in the 

 same time will be the same in both orbit*, since the 

 mere revolution of an orbit cannot affect the distance 

 of any point in it from the centre. The rate of ap- 

 proach being the same, but the angular velocity being 

 different, it is evident, that the centripetal force must 

 be different; less, for instance, where the angular ve- 

 locity is greater. The approach made at any point in 

 an instant in either orbit will be the excess of the cen- 

 ; tripetal force above what would be necessary to retain 



VOL. VIIL PAJBT I. 



the body in a circle moving with the same angular ve- Application, 

 locity ; hence the excess in the one orbit will be equal S ^*"Y"""'' 

 to the excess in the other. Let a, b denote the centri- 

 petal forces in the compound and simple orbit ; c, d 

 those in the circles described with the same angular ve- 

 locities. It appears that a c=A d, hence transposing 

 a b-=c d ; but in each orbit the centripetal force ne- 

 cessary to describe a circle with the same angular velo- 

 city is inversely as the cube of the distance from the 

 centre, (Prop. 13.) and therefore the difference of cen- 

 tripetal forces necessary to do the same must also be in- 

 versely aa the cube of the distance. Hence, at any 

 point the difference of centripetal forces, employed in 

 describing the orbits themselves, will be inversely, as 

 the cube of the distance. 



PKOP. XVII. 



If there be two free bodies, the one cannot remain 

 at rest, while, by its attraction, it causes the other to 

 move round it ; but if the two bodies receive equal im- 

 pulses in opposite and parallel directions, their centre 

 of gravity will remain at rest, and they will describe 

 similar curves. 



The first part of the proposition is manifest from the 

 third law of motion, for as the one body attracts the 

 other, the other will attract the first, and cause it to 

 approach. 



The second part will appear thus : Let A and B be PIATE 

 the two bodies, C their centre of gravity. It follows, CCXLI. 

 (see GRAVITY,) that AC : CB : : the mass of B : the F* !* 

 mass of A, or shortly : : B : A. Let the bodies receive 

 equal and parallel impulses in the directions BF and 

 AG, and suppose that (leaving out the attraction) the 

 body A woulil move along AG in a moment ; join GC, 

 and produce it to F ; BF will be the line past over by 

 B in the same moment ; for the impulses being equal, 

 the velocities will be inversely as the masses, that is, 

 directly as AC : BC, but by similar triangles AG : BF : : 

 AC : BC. Again, suppose that, in consequence of the 

 mutual attraction, the bodies, in the moment alluded to, 

 describe the curves BD, AE, then GE, FD will be the 

 momentary deflections 



The attractions being equal, GE will be to FD : : 

 B : A, that is, AC : CB : : GC : CF ; and hence the re- 

 mainder EC will be to the remainder CD also in the 

 same proportion, viz. : : B : A. Hence the same point 

 C will still be the centre of gravity of the two bodies 

 when they have arrived at E and D, and hence it will 

 be so continually. 



Again, it is clear that the small arches AE and BD 

 are similar, since all the straight lines connected with 

 the one are proportional to the corresponding lines con- 

 nected with the other. The arches descril>ed the next 

 moment will be similar, for a like reason ; and hence 

 the whole arches described in equal finite times will be 

 similar. 



PROP. XVIII. 



The same supposition being made as in last proposi. 

 tion, the curves described will be similar to tne curve 

 which one of the bodies would appear to describe to a 

 spectator situated on the other, and conceiving himself 

 at rest 



Suppose the spectator at A : Suppose that the Fig. It, 

 arches AE and BD are described in a moment, and 

 may therefore be considered as straight lines. By the 

 principles of apparent motion, the body B will appear 

 to move in the direction BD with a vel.=: the sum of 

 the two velocities, viz. BD, AE. ilcucc, producing 

 3o 



