—.> 
. of Central Forces. ° 371 
CP=P, Cn==y, Cp=p, the velocity at V==v, 
the velocity at »==v, the velocities being mea- 
fured by the {paces defcribed in the time (1). 
If O be the centre of curvature of the trajec- 
tory at the point #, then it is well known that 
On= Ha Draw, OR perperidicular to Cz; the 
triangles Cpn, OR» are fimilar, therefore’ Cu 
() Cp (e) 2: On (2): Rae Yay chord of 
curvature’ pafling through the centre of force. 
But the centripetal force, .eftimated by the velo- 
city generated in the time @), is = the ‘fquare 
of ne velocity divided by 4 the chord. of cur- 
yo Stee uv \ 
vature =v? farsi L_VEb 1 and p*: Pe: U3 U3 
Se Es ) 
: . ' : P? yp? 
therefore the centripetal force = < - oS eet 
Y bora } 
this expreffion be put = ——., 4 being a con- 
. Pp es ons 
ftant quantity ; wherefore wt om Gry: 
P 9 i 
Multiply both fides of the equation by 5, and 
P?y* ~ Artn 
take the fluents, then £2 = ORES Mt DSSS 
2p m—1l xX y m—1 
P?y? 
But at V this equation becomes 
Att} 
n—1 + r™ 
P?y? v" at ns and yn—1 _yn—t 
F ¢ — = TR OR Sy 
2 p* 2 oat asi eee yr b 
Q" Pay 
, therefore the correct fluent. gives 
