98 Dr. Waring on 
The five equations are eafily reduced to three by extermi- 
nating the quantities v and y'. 
The fiuxional equation refulting will moll: commonly be of 
the fifth order, as evidently appears from the nature of the 
problem. 
2 . The fame principles may be applied to determine the 
curves, when the bodies move in mediums, of which the 
refiftances are given : for example, fuppofe the refifiances to 
vary as a function of the difiance from a given point into a 
function of the velocity : to the forces in the directions of the 
tangents contained in ti^ie preceding cafe mufi be added or fub- 
tradted the given refifiances for the forces in the directions of 
the tangents, and the remaining procefs will be the lame as is 
before given. 
If two bodies defcribe fimilar orbits round a common center, 
either quiefcentor moving uniformly in a right line ; the forces 
and velocities and refifiances of the medium will be to each 
other in correfpondent points as their refpedtive diftances from 
the center. 
PROP. XIII. 
Given the forces adting on any bodies, and tending to points 
cither moveable or quiefcent, or in the direction of the tan- 
gents, &c. ; to find the curve defcribed by one of the bodies. 
* i. Afiume .v andjy for the abfcifs and ordinate of the curve 
required, and from thence may be deduced the difiances from 
any quiefcent center of force, and confequently the force j in 
that direction ; refolve it into two others, one in the direction 
of the tangent, and the other in a different one ; for example, 
let it be in a direction perpendicular to the tangent, and from 
their fluxions x and y , and the force f may, by the method 
3 before 
