^436 PHILOSOPHICAL TRANSACTIONS. [aNNO: 1 /OS', 



Theorem. If a body, urged by a centripetal force, move in any curve; 

 then, in every point of the curve, that force will be in a ratio, compounded of 

 the direct ratio of the body's distance from the centre of force, and the reci- 

 procal ratio of the cube of the perpendicular on the tangent to the same point 

 of the curve, drawn into the radius of curvature of the same point. 



Demonstr. Let qao, fig. 1, pi. xi. be any curve, described by a moving 

 body, urged by a centripetal force tending to the point s. And let ao be an 

 arc described in any very small time, pm its tangent, ar the radius of the circle 

 of equal curvature, that is, of which the element of the periphery coincides 

 with the arc ao. And let sp be a perpendicular on the tangent ; also draw om 

 and ON parallel to sa and sp. Let om express the force by which the body 

 at A is urged towards s : then the force by which the body recedes perpendicu- 

 larly from the tangent, will be as on ; that is, the force tending towards r, and 

 causing the body, moving with the same velocity as before, to describe a circle 

 equicurved with the arc ao, will be to the force tending towards s, by which 

 the body moves in the curve ao, as on to om, or, by equiangular triangles, as 

 SP to sa. But the centripetal forces of bodies,.. moving in circles, are as the 

 squares of the velocities applied to the radii, by the Cor. to Theor. 4, Newton's 

 Principia ; and the velocity is reciprocally as sp ; therefore the force on, or the 



force by which the body can move in an equicurved circle, will be as — ; -i 



^ -^ • . sp* X AR 



but it has been shown that sp is to sa, as the force tending towards », by which 

 the body can move in an equicurved circle, is to the force tending towards s ; 



and the force tending towards r is as — j ; therefore as sp : sa :: 



SF X A R 



which the;refore is as the force tending towards s. q. e. d. 



SP* X AK * SP^ X A a' 



CoroL If the curve aAo be a circle,.. the centripetal force tending towards 



s, will' be as ^, fig. 2. Therefore if the centripetal force tend towards s, 



a point situated in the circumference, then, by Eucl. 32, iii, the angle pas = 

 the angle at aqs ; therefore by the similar triangles asp, asq, it will be aq : 

 AS :: as : 9p ; therefore sp :^ 



~, and 8P^ =t — ,; hence ^ = - ^ ^^^ = ^ ; that is, because aq is given, 



the force will be reciprocally as as^. 



Let DAB (fig. 3) bfe an ellipsis, whose axis is db, its foci p and s, also 

 AR and OR two perpendiculars to the curve very near together : draw kl 

 and OT perpendiculars on sa, and km perpendicular to or. Then because 

 SA : SK :: (by Eucl. 3, vi) pa -f sa : fs, that is in a given ratio, the 

 fluxions of SA, SK, that is at, kA, will be proportional to sa, sk : also, 

 by Conic Sections, al = half the latus rectum = ^l: further, because ka 



