j6 Dr. Waring on 
? chord of the circle of curvature, which pafies through S, 
write C ; and for PO x (— / v x x r d=&c.) fubftitute 
8 F O 1 
PI, and for — write B ; and for =±r/' / x -±±j" x -f &C.. 
sr* J s p s p 
fubftitute D, and the two preceding equations become v z — 
/xC + H and — vv = (B/T- D) A, where A denotes as be- 
fore the increment of the arc of the curve : from the firffc 
equation vv — 
fx C + Cf+H_ 
= - (B/’A + DA ) and confequently 
cyq- (C + 2B A) f+H + 2DA = 0, from which fluxional equa- 
tion may be deduced the force f tending to the center (S) 
rr - C — 1 x e 
r • • f -- 
J c x j (H -j- 2DA) x e J c , where e is the 
number, whofe hyper, log. = 1. 
Cor. Fig. 1. Lety v , j'\ f"\ &c. be each = 0 , then will 
r 2 b\ 
D — 0, PI — 0, and confequently y"(2DA + H)x^‘' c = 
/ 2 BA 2BxSjX 
C ~ CXpy, 
XPO 2Sy 
_ 
_I x e J s ->’ ; 
whence f=z ( 
as is generally known r 
aC " 7 ^“S/xC 
where a denotes an invariable quantity. 
Cor. The force f being found, the fquare of the velocity 
may be deduced from the equation v z =f xC + H, and the 
.a ■ 
time from the fluent of the fluxion — — — — 
v vy x c+H 
2. Let the body move in a curve of double curvature, and 
let the forces , &c. tending to all the points M /7 , M" 7 , 
&c. (except two, M and M 7 ) be given ; to And the forces 
tending to the points M and M 7 * 
Afliime 
