Centripetal Forces, $ r 
D'% D'", &c. from the fluents of the flux!ons/D,y v D / , &c. ; 
and confequently in terms of D and D r , which let be Z, then 
will Z = — ; but v'= -aZ = PO x Ifx X — + f» 
gz/y/Z gz/zyzzz 
x ^.TTp- =i=/ /7/ x — &c.) a fluxional equation of the fecond 
order exprefling the relation between D and D', and their 
fluxions. 
2 . To find an equation exprefling the relation between 
a' = SM and y— MP, where SM (x) is the abl'cifs beginning 
from S and continued in the line SS', and MP ( y) the per- 
pendicular ordinate of the curve deferibed by a body added on 
by the above mentioned forces : in the fluxional equation 
found before for D and D' and their fluxions fubfiitute 
(#* + y 2 )* and ((SS'zt#) 2 +y 2 )* and their fluxions, and there 
refults the equation fought. 
Cor. It eafily appears, that the general fluent may contain 
two invariable quantities to be aflumed at will, or according to 
the conditions of the problem ; that is, at a given difiance the 
velocity and the direction may be aflumed at will, and confe- 
quently the general fluxional equation expreffing the above 
mentioned relation will be of the fecond order, if no fluents 
are contained in it. 
Cor. From Py and Py', and the points- S and S' being given, 
can eafily be deduced geometrically the diredhon of the tan- 
gent and the lines Sy, §/, &c. ; for divide the line SS' in r, 
fo that Py — Py' : SS' :: Py : Sr, and through r draw the line 
Pr, the perpendicular to Pr through P will be the tangent 
yPy'; to this line the perpendiculars from S and S' wilt be the 
lines Sy and S '/ required. 
Vol. LXXVII1. M Cor. 
