Central Forces. 345 
of T=F’sin.) must be changed, when c’? is calculated by (2). If 
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- 
, ti a.Va, iE ids 
ion for EF” is easily found by (E), for — g7--—Go=F”, or VS. 
OFdr+d.V2) ie 
feed — ), also by (EF) (noe .d.V? =d(RE”), hence 
(2F”dr+d(RFE”)) 
Qds 
»(F). Prin. B. 1, prop. 17; Vince’s Flux- 
2F’dr+d(RE” 
ions, prop.43. If F¥=V?2h=F’”RA, then = ote : 
Wess 1 
(G), in which h=the density of the medium; Prin. B. II, prop. 18. 
If the centre of force is supposed to be removed to an infinite dis- 
tance, so that r may be supposed to move parallel to itself, and if 
F”= const. let x, z be the abscissa and ordinate of the curve, z be- 
ing parallel to 7 and w perpendicular to it; let 2 and 2 have their ori- 
: ; ; dx? +dz? 
gin at the highest point of the curve, then dr= — dz, LS ee ae 
Qdz(d?z)?—ds*d°z Savi 
make dx constant, then dR=—— (ez)? ; by substituting 
. : By oil ads 
these values of dr, dR in (F), (G), they become FY 9(de2)”” (H), 
3 
esac, (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 act upon it to be reduced orthographically 
to the fixed plane 
