Centripetal Forces. 69 
between S p' and S y' (a perpendicular from the point S to p'v , a 
line touching the curve in the point p'} ; in which two curves 
PP 7 L and pp'l , the forces and velocities at any equal difhnnces 
SP and S p are equal, and confequently the perpendiculars SY 
and Sy, at the above-mentioned equal diftances SP and S p are 
to each other in a given ratio N : n. 
In the equation exprefling the relation between SP / and SY' 
for SP / and SY 7 write refpedively S p and 
S/xN 
and there re- 
fults the equation fought : for the diftances SP and S \p' being 
equal, the perpendiculars SY 7 and S \y' are as N : n. 
Ex. 1. Let S be the focus of a conic fedion, then will 
| C 2 x - D — , SY : — P 2 , where T and C denote its tranfverfe 
and conjugate axes, and D the diftance SP ; for P write 
— x p 9 and there refults the equation ?C 2 x -^- = ^ x p\. 
which is an equation to a conic fedlion of the fame name (yh m 
ellipfe, parabola, or hyperbola) as the given curve, of which* 
the tranfverfe axis is T, and conjugate = and perpendi- 
cular from the focus to the tangent =p. If T and C are infi- 
nite, and confequently the curve a parabola, and the equation 
|LxD = P, then will the l at us return of. the refulting equa- 
. L x n z 
tion be • 
jN 
Ex. 2. Let S be the center of the logarithmic fpiral, then 
will the equation be a x SP = a x D = SY = P, and confequently 
the refulting equation a x D = x/>, whence ^ x D = p an 
equation to a logarithmic i pi ml having tne fauio cento 1 .. 
Ex. 3. Let T and C be the femi-conjugate axes of a conic 
fedion, and S its center ; then will the equation exprefting 
the 
