1 86 
Dr. Waring on the 
PROBLEM I. 
Fig. 2. Giv r en the law of attraction of each of the parts of 
a given line in terms of their diftance from a given point P ; 
to find the attraction of the whole line ab on the point P. 
Find the attraction of the line ab on the point Pin the two 
directions Vf and fb by the following method. Draw Vx from 
the point P to any point x of the line ab, the force acting on 
the point P by the particle xy will be the given function (de- 
termined from the given law of attraction) of the diftance into 
the particle ; draw alfo P h perpendicular from the point P to 
the line ab, and let Vf — a, hj — b , and fx =y ; then will the 
diftance Fx= s /(a 2 dti2by-]-y~), and the function of the dif- 
tance into the particle xy = <p (^(^zt^iy+y*)) Xy — F (y) xy; 
let this be denoted by lx fituated in the line P^, which refolve 
into two others nx'zn- — fituated in the line ab % 
Yxzz \/ (a ±:2by-\-y ) 9 
and In (in a direction parallel to Vf} — 9 
fluents of the fluxions 'VJL Lf jl an j ^ - x Z i il l contained be- 
irx Yx 
tween the values af and fb of the line fxzzy, which fuppofe 
Y and V refpeCtively ; through the point p draw Py parallel to 
fb~Y, and in the line Vf affume P^==V; complete the 
parallelogram P uzy ; P z will be the force of the line ab on 
the point P. 
Cor . If F : (y) varies as any power or root (2 n) of the dif- 
tance Vx ~ zby-\-y z }, and ft — § be an integer affirma- 
tive number or o, the fluents Y and V of both the fluxions can 
be found in finite algebraical terms of y ; if ft — § be an integer 
negative number, both the fluents can be found in the above- 
1 mentioned 
