VOL. LXXIX.] PHILOSOPHICAL TRANSACTIONS. 5/5 



points E and e of the curve draw ed and ed perpendicular to the axis ab, and es 

 perpendicular to the arc Ee of the given curve at the point e, and meeting the 

 axis ab in s : then will the evanescent solid Ewfe =j&XpeX fd X f/= p X 



FD X PS X Drf (because p/= — -^—) = p X {\/ (z^ + f) — z) X (zs^+yi), 



where z and y denote respectively the absciss pd, and its correspondent ordinate 

 DE of the given curve. 



The increment of the attraction of the surface ef on the point p, in the direc- 

 tion PD, will be as the increment of the surface ( 6 X pe x Drf) x — X force of 



'^ PE 



each particle = p X pd X d</ X given force of the particle ; but the fluent of 

 the fluxion pd X T>d contained between the points e and p is = ipE"^ — ^pd'^ = 

 ^ed^ ; whence the attraction of the evanescent solid EP/e is as -^p x ed^ x 

 FfXF:^x^-{- y^i force of each given particle at the distance (pe = '^{x^J^y*\\ 



PS _^_^___ gy. -f- %jy 



=? 4-p X ED^ X 71 X Drf X P : sfz^ + 3/' = W X -^(^J^ X p : >/(z'^ + 

 3^*) ; the fluent of which, properly corrected, is as the attraction of the solid on 

 the point p ; p denoting the circumference of a circle whose radius is 1 . 



Cor. 1 . The fluxion of this solid is i j&j/^i = y, which deduced from the pre- 

 ceding principles —p X (-/(z^ + 3/^) — z) X (zs- + yij) = v, and consequent- 

 ly their fluents between two values of z, which correspond to two values of^=o, 

 will be equal to each other. 



Cor. 2. The increment of the attraction of this solid, as given in this proposi- 

 tion, \p X y'' X '^y^^jr^ X F : '/(z^ -h / = uj but in the preceding pro- 

 position the force of a circle on the point p = p x f m X f : m, where u = 

 ^{fi + ?/^), and fl =: z, and y or u the only variable quantity contained in 

 the fluxion ; consequently the fluxion of the attraction of the solid p X z 



/z -7-" X P : (z^ + yy^ = wj therefore, if for the fluent of -rrrv-^r X 



f : (z^ -f 3/^)' be substituted its fluent contained between the values a and the 

 value of y, which in the given equation corresponds to z ; then the fluents of 

 V and Mr, contained between the 2 values of z which corresponds to 2 values of 

 r/ = 0, will be equal to each other. 



The difference of the fluents of y and v, &c. contained between any other 2 

 values of z, can easily be deduced from the difference of 2 segments of spheres. 



1. It may not be improper to remark in this place, that from different methods 

 of finding the sum of quantities, the fluents of fluxions, the integrals of incre- 

 ipents, &c. quantities may often be deduced equal, which otherwise cannot with 

 out some difficulty j of which instances are contained in the Meditationes, and I 



