jo Dr . Waring on 
the relation between the diftance D and perpendioular P be 
D'zt--— = T 2 r±rC 2 ; for P write as before^-, and there re- 
P* n 
j.2 T 2 C 2 a 
fults the equation D 2 — y~3^=T 2 z±rC% an equation to a conic 
fedtion of the fame name, of which the tranfverfe and conju- 
gate diameters are refpedtively two roots (#) of the equation 
afrfc— 1 -if =rT 2 — C% becaufe in this cafe p — D. 
N x 1 
The fum or difference of the fquares of the tranfverfe and 
conjugate diameters, in all the refulting equations, will be the 
fame. 
Cor. In every equal diftance, the chord of curvature pafting 
through the center of force is the fame; for the forces iu that 
direction, and the velocities at every equal altitude are the fame. 
$ 
PROP. III. 
i. Fig. 4. and 3. Given an equation A: =<?, exprefling the 
relation between the abfcifs SM=J? and ordinate MP zzy ; to 
find the equation exprefling the relation between SP — s/ x~ 4- y 1 
and SY = P, the perpendicular from S on the tangent PY. 
From the equation A — 0 find x = B y, which fubftitute for x 
in the equation (x z +y 2 )i x P — xy±zxy deduced from the fimilar 
triangles P lo, MTP, and STY, where lo—x and P o=y; let 
the refulting equation be C = 0; reduce the three equations 
A—o , C = o, and Y +/ - SP 2 = D 2 into one, fo that the un- 
known quantities x andy may be exterminated, and there re- 
fults an equation exprefling the relation between D and P. 
Cor. Hence from the equation expreffing the relation be- 
tween x and y, the abfcifs and ordinate of a curve, can be de- 
duced an equation exprefling the relation between the diftance 
SP and perpendicular SY 5 and from the equation exprefling 
5 the 
