574 PHILOSOPHICAL TRANSACTIONS. [aNNO I789. 



(3/) X i X ~{^] X V and i X ^^,^. ±11^ , + ,.y = y X ^^, contained be- 

 tween the values of y, which correspond to the extreme values of x, which sup- 

 pose Y^ v', and z; and draw through the point p the lines pz/ and Pz respectively 

 parallel to the ordinates pm and to the absciss aj&, and equal to r x y' and v'; 

 assume pw in the line (pa) = t X z, r and t denoting the sines of the angles, 

 which the ordinates pm and line ap make with the absciss xp : reduce these 3 

 forces VT/, pz, and pm, to one f/; thence p/"will be the force of the surface on 

 the point p. 



Cor. 1 . If for y and i/ be substituted their values in terms of x and i, de- 

 duced from the equation expressing the relation between the absciss a/j and ordi- 

 nate pm of the given curve, thence will be deduced the above-mentioned fluents 

 Y, V, y', v', and z, in terms of x: and in the same manner, if for x and x be 

 substituted in the fluxions or fluents resulting their values i/ (u^ — a'^ _|_ ^a ^"2^ 

 + sa% and its fluxion, there will result the above-mentioned fluxions or fluents 

 in terms of u the distance from the point p. 



Cor. 2. Let the curve be a circle, of which a is the centre, pa a line per- 

 pendicular to the plane of the circle, and the ordinate pm perpendicular to the 

 absciss a/>; the forces on each side of the absciss Ap will be equal, and the force 

 in the direction of the absciss Ap will be equal to that in the contrary direction j 



the force in the direction (pa) = 4 xy ^/^2 _ ^.n Xj-^jr=-=.^ x f : -v/ (m^ -}- 3/^) 



= w, in which f : . ^«» + y-^ is the function of the distance, according to which 



the given force on the particles varies; the fluent /^ "-^ X p : V(m^ -f- y"^) 



is contained between the values O and i/ (r* -\- a^ —- u^) of the quantity y, and 

 the fluent w is contained between the values a and */ {a^ -}- '*) of the quantity 

 Uj where a = pa and r the radius of the circle; but the same force is = 2 X 



3.14159 &c. X fau X F : M, where p : u denotes the given function of the dis- 

 tance w, and the fluent is contained between the values a and y^oM^ of u. 



Prob. 3. To find the attraction of a given solid on a given point p. Find 

 the attraction of every parallel section on that point by the preceding problem, and 

 multiply it into the correspondent fluxion of the first abscissa ap; and also find 

 the fluent of the resulting fluxion, which, properly corrected, multiply into the 

 sine of the angle which the first abscissa makes with the parallel sections, and the 

 product will be proportional to the attraction of the solid on the given point p. 



1. Fig. 8. Let the solid abch be generated by the rotation of a given curve 

 round its axis ab, which passes through the point attracted p, and this solid be 

 supposed to consist of small evanescent solids, whose bases are the surfaces ep, 

 ef, &c. of spheres of which the centre is p, and altitudes p/, &c. the increments 

 of the base ab contained between the 2 contiguous surfaces ep and efi from the 



