. Central Forces. a6 
of T=F’sin.. must be changed, when ¢’? is calculated by (2). 
the particle is supposed to describe a given curve around a given 
centre of force, and the expression for F” is given, then the express- 
? 3 
é 
; : ; dV ds 
ion for F” is easily found by a for — dr yp or F’= 
ateaee: +d.V?) 
3 also by (E) ve=F’ ..d.V? =d(RF”), hence 
2Fd (RE” 
i= _@Frartane (F). Prin. B. II, prop.17; Vince’s Flux- 
(2F"dr-+-d(RE”)) 
oI Rar 
. lons, prop. 43... If F’-=V2h=F’RA, then h= — 
(G), in which k= the density of the medium; Prin. B. II, prop. 18. 
If the centre of force is supposed to be removed to dn infinite dis- 
tance, so that r may be supposed to move parallel to itself, and if 
F”= const. let v, z be the abscissa and ordinate of the curve, z be- 
ing parallel to r and x perpendicular to it; let « and z have their ori- 
. : dx? +dz?, 
gin at the highest point of the curve, then dr= —dz, R=——7,-—3 
Qdz(d?z)* —ds?*d%z es 
make dx constant, then dR= = = - 3 by substituting 
3zds 
F 
these values of dr, dR in (F), (G), they become Faz)” (H), 
3 
epee (I). Prin. B. II, prop. 10; Méc. Cél. Vol. I, p. 26; 
also Vince’s Fluxions, at the place cited above. It may be observed 
that (D) are applicable when the disturbing force is not in the plane 
of the curve described by the particle around the first centre, sup- 
posing r to be changed to 7’, F to P; that is, by supposing the par- 
ticle and the forces which ak upon it 46 be reduced orthographically 
to the fixed plane _ 
