84 Dr. Waring on 
nate, if fituated in different planes ; from the centers, &c. 
and forces given, find the fum F of the forces in any direc- 
tion (PL) (the direction of the tangent excepted) adting oil 
the body at the point P, and the chord of curvature C of the 
curve at the fame point and in the fame direction ; in the equa- 
tion v~ = | F x C for v z fubflitute the value found before, and 
there refuits an equation expreffing the relation between the 
diftances from two points, or an abfci'fs and ordinate, &c if 
the forces adt in the fame plane : but if the forces aft in dif- 
ferent planes, find the fum F and F 7 of the forces at the point 
P in directions which are not both fituated in one plane with 
the tangent and each other ; and alfo the chords C and C 7 of 
curvature in thofe directions in terms of the diftances from 
three points, or two abfciffe and one ordinate, &c. In the 
equations f=:lFxC and v~ — \ F 7 x C 7 for v z fubflitute its 
value found from the principles before given; and there refult 
two fluxional equations of the fecond order expreffing the rela- 
tion between the diftances from three points, or two abfciffe 
and an ordinate, &c. 
PROP. VIII. 
Fig. 9. Let a body move in 4 a curve P p, See. and be adted 
on at P 7 by a force f (which is as any function of the 
diflance SP 7 ) tending to S ; let the velocities at P and p be 
reprefented by the lines YP and yp in the direction of the tan- 
gents to the points P and p ; refolve thefe forces YP andyjfr 
into two others Yk and £P, and yl and Ip , of which one £Y 
and yl is parallel to the line SL ; the other £P and Ip is pa- 
rallel toMP: let a body fall in the right line LS, and the 
force adting on the body at M 7 be to the force adting on the 
body moving in the curve at P 7 :: SM 7 ; SP 7 , and P 7 M 7 , PM 
and 
