Dr. Waring on the 
relation between the abfcifs and ordinates of the given curve, 
find the abfcifs in terms of the ordinates ( pm) — ^ : (y), and 
thence x~q> : (jy) xy and v /(^z /2 =i =2 sa y x x 2 )=: cp' : (jy), where 
PA — a' and j=: cofine of the angle, which the abfcifs Ap 
makes with the line PA ; then find the fluents of the three 
Vxa: 
fluxions xxY =y x Y x 9 : (jy), x x = <P : ( y) X 
a'V . a'V 
y X - ! \ y \ X V and x X 
V ( a' z ±: 2 sax-\-x 7 ') 
= y X “t — 7-rContained be- 
tween the values of y 9 which correfpond to the extreme 
values of x 9 which fuppofe Y\ V 7 , and Z ; and draw through 
the point P the lines Pjy and Pz refpeftively parallel to the 
ordinates pm and to the abfcifs Ap and equal to r x Y / and V 7 ; 
affume P u in the line (PA)=/xZ, r and t denoting the fines 
of the angles, which the ordinates/^ and line AP make with 
the abfcifs Ap : reduce thefe three forces Pjy, P z 9 and P u 9 to 
one P f 9 and Pf will be the force of the furface on the 
point P. 
Cor. i. If forjy andj) be fubftituted their values in terms of 
and x 9 deduced from the equation expreffing the relation 
between the abfcifs Ap and ordinate pm of the given curve, 
thence will be deduced the above-mentioned fluents Y, V, Y r , 
V 7 , and Z, in terms of x ; and in the fame manner, if for x and 
x be fubftituted in the fluxions or fluents refulting their values 
^/(p — + i~a /2 ^ = f z sa / 9 and its fluxion, there will refult the 
above-mentioned fluxions or fluents in terms of u the diftance 
from the point P. 
Cor. 2 . Let the curve be a circle, of which A is the center, 
PA a line perpendicular to the plane of the circle, and the 
ordinate pm perpendicular to the abfcifs Ap ; the forces on each 
fide of the abfcifs Ap will be equal, and the force in the direc- 
6 tion 
