344 Central Forces. 
the place of the particle, estimating the chord on r. Then, since 
dr 2 : / 
cdc =Tr2dz, =F, = cole (7) is easily changed to the forms 
c/2 a( dr2 cdeédr c’? 
all 9 4 2 # 
Laaia 
r3 
; 6) ies 
seid |e 5 s — Tcot.p= 
ae Page ne ; 
2dr 2dr 
ORK 1 2 1 cf? 
— 54 (aga ay) Teotb=— zd 55) - Toot =aeg —T cot. 
dr dr 
V2 V2dp V2 c2 ef? 
cn —Tcot.)= R ~ Peet. $= peR Tot. v= Renee Sak 
Tees gee idee ge ea ea 
“gr aaa gaa r?sin.?-Ldr his Pa) 
2dr 2dr 
T oe d.V? pal 
sae Eco - a(e +Ttan.l=- Qar Psin.beose) 
Qdr 
d.V? , 
T cot.p= — 97, + Ttan.p=F, (D).. 
If the disturbing force acts constantly in the direction of the ele- 
ments of s and is denoted by I’, and if IF” denotes the attraction 
towards the first centre, then F=F”—F’cos.), T=F’sin.lb3 by 
substituting these values of F and T in (D), since —F’ cos. is 
42 
found on both sides of the resulting equations, they become | — 
Ce Gig? Nu 100) 2b $004 ental ce 1 Ce 
34(saes) p73 (Se) =~ 54a) =— 2p) 
dr dr dr 
Aaa NCE PINE cr cr cc | 
7, podr es, par oR pe R = "Rant =—? (=p) 
Qdr 
ede! yor? uM d.V2 1 dN? as 
Senter ete cos. mb Qdr heosb >. Bar dee 
2dr 
=F”, (E). It is evident that the formule (E) will apply whether 
the force F’ is accelerating or retarding; for instance, they are ap-' 
plicable when the particle moves in a resisting medium, in which 
case the particle is retarded in the direction of the elements of s, and 
as EF” is a retarding force its sign must be changed; also, the sign 
