[ 6 ; ] 
VII. On Centripetal Forces. By Edward Waring, M. D. 
F. R. S. Profejfor of Mathematics at Carr bridge. 
Read January io, i y S 3 • 
PROP. I. 
i* T" ET a curve P/>N (Tab. II. fig. 1.), of which the per- 
J — J pendiculars to the two neared: points P and p of the 
curve are PO and pO , and confequently O the center of a 
circle, which has the fame curvature as the given curve in the 
point P; draw PY and ly tangents to the curve in the points P 
and p ; from S draw Sy and S£Y refpedtively perpendiculars to 
the tangents ly and PY ; and let S-6Y cut the tangent ly in h ; 
then will ultimately hY ( - P) be the decrement of the perpen- 
dicular SY = P ; and the triangles lb Y and PO/> be fimilar : for 
the angles PO/> and hlY are equal, and the angles lYh and 
OP^ right ones; therefore PO : P p :: IY ultimately = PY : Yh 
decrement of the perpendicular, whence 
1.2. Fig. 2. and i. The force in the direction PS is as the 
ultimate ratio of 2 x QR (the fpace through which a body 
is drawn from the direction of its motion in the tangent in a 
given time towards the center of force) ; but ultimately 2QR = 
2 -~~> where QP is as the fpace deferibed in a given time, and 
confequently as the velocity (V) of the body at the given 
point P, and PV the chord of curvature in the dire&ion SP. 
K 2 1.3. 
p?= 
Y b x PO Y h x PO 
IX 
PY 
