' the time of describing the whole of it is called the pc- 

 """" """*' riorlic lime. 



The following are the most important propositions on 

 the subject of central forces. 



PROP. I. 



If the centre of attraction remains always the same, 

 the curve will lie wholly in one plane, passing through 

 that centre ; and the areas described by the radius vec- 

 tor, will be proportional to the times of description. 



First, suppose the central force to act during equal 

 fi"'.. fm * te . lntervals of time; suppose c the centre of at- 

 traction, AB the line passed over in one of the equal 

 intervals, the body with its uniform motion would, 

 during the next equal portion of time, go over a line 

 BD=AB, but at B it is acted on by the central force. 

 Suppose the momentary action is such, that in the same 

 time the body would move along BE, then comple- 

 ting the parallelogram, BF will be the real line of the 

 motion. Joining CD, the A ABC is = A BDC, be- 

 cause they have equal bases, and the same altitude 

 and the AFBC is = A BDC, because they are on the 

 simp base, and between the same parallels ; therefore, 

 ABC=FBC. The same thing may be shewn with re- 

 gard to the next triangle, &c. Hence, the sum of all 

 the triangles, or the whole area, described in a given 

 time, will be proportional to the time of description. 

 Also, all the triangles are in one plane ; for CFB is in 

 the plane ofCDB, since they are between two parallels 

 which necessarily lie in one plane, and CDI5 is evidently 

 in the plane of ABC; then-fort- CFB is in the plane 

 of ABC. It in clear, also, that the point C must al- 

 ways be in the plane of the Figure. Supjwse now that 

 the central force acts incessantly, or that the equal in- 

 tervals of time are infinitely .small, the path then be- 

 comes a curve, and the triangles become infinitely small 

 but still the same demonstration will hold; and hence 

 we infer the proposition. 



Conversely. PROP. II. 



If the curve described lie wholly in one plane, and 

 the radius vector drawn from a certain point in the 

 lane, always describe around that point areas propor- 

 tional to the times, that point is the centre of attrac- 

 tion. 



For around any other point than the centre of at- 

 traction, the areas described in equal times cannot be 

 equal. 



P* T . hus > in , the * a "ie Figure, take any other point 

 G, it is evident that GBF cannot be equal to GBA 

 for ,n order that tin's might hold, DF would need to 

 >e parallel to GB, whereas it is parallel to CB 



DYNAMICS. 



295 



stant, and therefore the angle C the angular velocity Application. 

 when the body is at A. Let AD be a circular arch, of '""V"' 

 which C is the centre. It may be regarded as a per- 

 pendicular on CB. 



From the nature of die circle, the angle C is = -; 



but the area of the triangle described in an instant be- 

 ing always the same, the perpendicular will be inverse- 



ly as the base ; hence, AD = 7^-, that is = , be- 



' AC 

 cause AC is ultimately = CB ; hence the angle C == 



r 



t ' le an S a ^ ar velocity, and r the radius 



PROP. III. 



The projectile velocity at any point of the curve is 

 inversely as the perpendicular let fall on the tangent 

 at that point from the centre of attraction 



r or the small triangle described in an instant by 

 the radius vector being every where of the same area 



base must be inversely as its perpendicular; but the 

 base is the projectile velocity, and the perpendicular 



the base is just the perpendicular oil the tangent. 



PROP. IV. 



The angular velocity is inversely as the square of the 

 distance or of the radius vector. 



Let ACB be the small triangle described in an in- 



AC 1 



vector, then a == . 



PROP. V. 



The centripetal force, at any point, will be proper- PLATE 

 tional to the versed sine of the arch described in an in- CCXLI. 

 slant, reckoning the versed sine toward the centre of Fi 8- 10 - 

 attraction. 



For the small arch AD may be considered as the di- 

 agonal of the parallelogram EB, of which the side AB 

 expresses the effect of the projectile force, and AE that 

 ot tlie central force. 



Cor. 1. You may measure the centripetal force by 

 the velocity generated in an instant, and then you will 

 employ twice AE ; but the ratio is of course the same. 



Cor. 2. BD, which is equal to AE, is the instantane- 

 ous deflection from the tangent, and may also serve to 

 measure the centripetal force. 



Circle PROP. VI. 



Sectors of a circle being proportional to the arches 

 on which they stand, when the body describes the cir- 

 cumference uniformly, the radius vector will describe 

 the area uniformly ; hence the centre of the circle must 

 be the centre of attraction by Prop. 1. The converse 

 follows in a similar manner from Prop. 2. 



PROP. VII. 



If the body describe a circle, the centripetal force at 

 any point is directly as the square of the projectile ve- 

 locity, and inversely as that part of the radius vector 

 produced which the circle intercepts, often called the 

 deflective chord. 



Let AB be the arch described in an instant, BD II the Fie 11 

 tangent, BD the tangent, C the point of attraction, then 

 AD measures the centripetal force. 



Joining AB and BE, it is easy to see from similar 



triangles, that AD : AB : : AB : AE ; hence AD = * 



AE' 



But the chord AB is ultimately equal to the arch, and 

 therefore equal to the projectile velocity ; hence, the 



centripetal force = -. , veh ' = veL> . 



def. chord ' def. chord' 



PROP. VIII. 



If the body move uniformly in a circle, the centri- 

 petal force will be directly as the square of the proiec- 

 tile velocity, and inversely as the diameter. 



For the centre of the circle will be the centre of at- 

 traction (Prop. C.;, and hence the deflective chord will 

 be the diameter. Hence, the Prop, follows from the 



