Centripetal Forces, 9 1 
— ; for it may reft at any point, where the refifting force is 
always equal or greater than the accelerating force. 
Cor. 'Let n be the number of vibrations, then the diftance 
of the arc, to which it will afcend from the loweft point at n 
vibrations, will be A — 2 na ; if A - 2 na be not greater than 
2 a y it will never pafs the loweft point. 
Philofophical enquiries require fome corrections, which do 
not enter into mathematical calculus; for example, in fome 
cafes the calculus changes the quantities from negative to affir- 
mative, &c. when from philofophical confiderations they are not 
changed ; and, vice verja , they may be changed to affirmative, 
&c. on philofophical confiderations, when they are not changed 
from the calculus : and alfo a body may ftop, &c. from philo- 
fophical confiderations, as in the preceding example, when it 
does not follow from the algebraical calculus, &c. It is fur- 
ther to be obferved, that refiftances are always to be taken 
affirmatively. 
Ex. 2. Let the accelerating force be as the arc, that is, the 
diftance from the loweft point, and the refiftance as the velo- 
city; then will the fluxional equation (F — V) A = — vv be 
(ax - v) x = — vv 9 which is an homogeneous equation of the 
firft order : write in it zx for v 9 and its fluxion for v, and 
there- refults the equation (ax - zx) x x= - zx z z - z'xx, whence 
(a -z) x — -zxz-z*x and - = "'3 and thence log. #= 
- i log. (a -z + z 1 ) (W) cir. arc, whofe radius is 
and tangent (z-f)+B, whence can be found v = x% 9 
and from curvilinear areas i = ^ . 
N 2 
If 
