96 -Dr. Waring on 
\ 
3. If the velocities v , v\ v" , &c. at every point of the arcs 
tf, of y a" , See. of the ( n ) above mentioned curves A, A\ A'\ 
Sec. be given in terms of their arcs, abfciffae, or ordinates, 
&c. and the places in which the bodies are fituated at the 
fame time in the arcs b , b\ b" , Sec. of fome other curves B, B', 
B /x , &c. find the correfponding velocities V, V', V", Sec. at 
the fame time of the bodies in the curves B, B', B", &c. ; 
then make — = -r = — = &c. = — , or which is equal to it = 
b' b" 
__or=— = &c. From the fluents of the fluxional equations 
refulting properly corre&ed will be found the arcs a, a\ a'\ 
Sc c. deferibed by the bodies in the curves A, A', A", See . in 
the fame time as the correfpondent arcs b 9 b ' , b " Sec . ; and 
from thence, by the method given in the preceding cafe, may 
be deduced the forces. 
The fame principles may be applied to bodies moving in re- 
fitting mediums. 
PROP. XII. 
Given the law of the forces of two bodies a&ing on each 
other, to find the two curves by them deferibed. 
Fig. 12. Aflume x and jy for the abfeifs (AP) and ordinate 
(PM) of one curve, and z and u for the abfeifs (AP") and 
ordinate (P'M') of the other ; where the abfciffae AP and AP' 
begin 'from the fame point A, and are fituated in the fame line ; 
then will the diftance (DrrM'M) between the bodies = 
- - t X 
s / % — x + Uz±zy ) ; let the forces of the body placed at M on 
that at M', and of the body placed at M / on that at M vary 
as <p : (D) =F, and cp / : (D)r:F / ; and let Mf> = x and^w= y; 
then will cofine of the angle wMM / to radius (i)be-^i* x 
t> N J x) 
y 
