7 2 
T)r. Waring on 
P R O P. IV. 
I. Let a body move in a given curve PH (fig. 5.), of which 
the velocity (y) at any point P is given: and let the forces 
f'\ f"\ &c. tending to all the given centers S // , S 7// , &c. 
(except two S and S 7 ) be given ; to find the forces f and j ' 
tending to the two points S and S 7 . 
Draw a line PO perpendicular to the tangent yVy' ; and 
from the given centers S, S 7 , S 77 , &c. draw lines S/ and Sy, 
S / 1 / and S 7 y, S // l // and S 77 jy r/ , &c. perpendicular to the 
lines PO andyPy, &c. ; then will x ±zf' x ii ~f" x 
1 O PS l b 
vr 
¥$' 
,z± :<kc. (where PO is 
the radius of the circle having the 
fame curvature as the curve in the point P), and 
x — =±= &c. (where A denotes the*arc of the curve PH) ; 
iy 
from the data may be deduced all the quantities contained in 
the above mentioned two equations, except f and f'\ and con- 
fequently from the two given fimple equations be deduced the 
forces fought f and f\ 
2. Let the velocity of the body moving in the given curve 
• I’ V p y' Py' 
be fuppofed always uniform ; then f x -|=k f' x x 
lb lb lb 
&c. — 0. 
Ex. Let the curve HP/ be an ellipfe, and the two foci S and 
S 7 the centers of forces ; then will EL, but the angle 
SPy = S / Py, and confequently = and f—f'\ but fince 
7 - = f *%+f *% = if * and v = a, then will f = 
po J sp J sp J sp ’ J 2$y x PO 
be the force tending to each focus. 
In 
