Refolution of attractive Powers . jg^ 
find the fum of the fame quantity or quantities, and there will 
refult feriefes equal to each other : for inftance (fig. 5 ), if the 
time of falling down the arcs AC and BC and their interpola- 
tions from the principles delivered in the Meditationes Ana- 
lytics, of which the difference let be D ; find the difference 
between the times of a body’s falling through BC when it 
began to fall from A and from B by a feries proceeding accord- 
ing to the dimenfions of ABzzcf a fmail quantity; and find, 
by a feries of the fame kind, the time of falling through AB ; 
the fum of thefe two feriefes will be equal to D. Similar pro- 
pofitions may be deduced from fluxional equations. 
4. In fome cafes the ratios of the times of bodies falling 
through fome particular diffances to each other may be eafily 
known ; for inftance, let the force vary as the m — 1 power 
of the diftance (v), and a be the diftance from which the 
body began to fall, then the velocity varies as <y(a m 
and the increment of the time as — — ; but if the parts 
A/( a m - x m ) r 
of different curves are proportional, then will a , x, and x vary 
in the fame ratio as each other, and confequently the time 
through proportional parts of the diftance will vary as 
a ' 2 —HL ; and if the bodies be refilled likewife by a force which 
varies as the — — power of the velocities, then the times 
through proportional parts will vary as before, that is, as 
a where a denotes the proportional diftances from the 
points where the forces and refiftanees are equal. 
p R o- 
