Jtttraction 

 of Solids. 



Fig. r. 



Prop. IV. 



Plate L. 

 Fir. 2. 



90 



A TTRA I O N. 



But when x is very 6mall, or y inde 



finitely near to A, * becomes the diameter of the 

 x 



circle having the same curvature with ACB at A, 

 and when x vanishes, this value of , or X, be- 



X T 



I x~ 

 comes infinite, because of the divisor x T being i" that 

 case = 0. The diameter, tlvrefore, and the radius 

 of curvature at A are infinite. In other words, no 

 circle, having its centre in AB produced, and passing 

 through A, can be described with so great a radius, 

 but that, at the point A, it will be within the curve 

 of equal attraction. 



The solid of greatest attraction, then, at the extre- 

 mity of its axis, where the attracted particle is placed, 

 is exceedingly flat, approaching more nearly to a 

 plane than the superficies of any sphere can do, how- 

 ever great its radius. 



4. To find the radius of curvature at B, the other 

 extremity of the axis, since y % =a T x T x 1 , if we di- 



vide by"a x, we have ^ =r 



.'/ 



But at B, 



a x a x 

 when ax, or the abscissa reckoned from B vanishes, 



is the diameter of the circle having; the same 

 a x 



curvature with ACB in B. But when a x=0, or 



;i x, both the numerator and denominator of the 



fraction 



a T x T - 



vanish, so that its ultimate value 

 u a" 



does not appear. To remove this difficulty, let a x 



=2, or x=a z, then we have y^=:a T ( j a z) T 

 (a z)\ But when z is extremely small, its powers, 

 higher than the first, may be rejected; and therefore 



(fl z)WYl 0r =ffl T(l_|f, &c.) There. 



fore the equation to the curve becomes, in this case, 



y=aT X oT(l |y a*+2az=a l ^. a* 



+ 2az~-^az. 



- 2 



Hence , or the radius of curvature at B = - a. 

 2z o 



The curve, therefore, at B falls wholly without the 



circle BKA, described on the diameter AB, as its 



radius of curvature is greater. This is also evident 



from the construction. 



To find the force with which the solid above de- 

 fined attracts the particle A in the direction AB. 



Let b ( Fig. 2. ) be a point indefinitely near to B, 

 and let the curve Acb be described similar to ACB. 

 Through C draw CcD perpendicular to AB, and 

 suppose the figure thus constructed to revolve about 

 AB ; then each of the curves ACB, Acb will ge- 

 nerate a solid of greatest attraction ; and the ex- 

 cess of the one* of these solids above the other, will be 

 an indefinitely thin shell, the attraction of which is 



the variation of the attraction cf the solid ACB, Attraction 

 when it changes into Acb. ol Solld - 



Again, by the line DC, when it revolves along 

 with the rest of the figure about AB, a circle will be 

 described ; and by the part C c, a circular ring, on 

 which, if we suppose a solid of indefinitely small al- 

 titude to be constituted, it will make the element of 

 the solid shell AC c. Now the attraction exerted bv 

 this circular ring upon A, will be the same as if all 

 the matter of it were united in the point C, and the 

 same, therefore, as if it were all united in B. 



But the circular ring generated by Cc, is :r tt 

 (DC Dc')=2xDCxCc. Now iVCxCe is 

 the variation of y x , or DC, while DC passes into 



Dp, and the curve BCA into the curve be A; that 



i i 

 is 2 DC X Cc is the fluxion of y' t or of a r x r X s , 

 taken on the supposition that x is constant and a va- 



riabl 



e, viz 



Therefore the space gene- 



rated by C c =. -^-aJx^a. 



If this expression be multiplied by .r, we have the 



element of the shell = - T ** a x. . 



In order to have the solidity of the shell ACBic, 

 the above expression must be integrated relatively 

 to x, that is, supposing only x variable, and it is the 



-X aJx^ a+C But C=0, because the fluent 



vanishes when x vanishes, therefore the portion of the 



shell AC c=-j.T a Tfl, and when x=a, the whole 

 5 



shell = -r-a* a. 

 5 



Now, if the whole quantity of matter in the shell 



were united at B, its attractive force exerted on A, 



would be the same with that of the shell ; therefore 



4?r 

 the whole force of the shell = a. The same is 



5 



true for every other indefinitely thin shell into which 

 the solid may be supposed to be divided ; and there- 

 fore the whole attraction of the solid is equal to 



a, supposing a variable, that is =: z- a. 



Hence we may compare the attraction of this solid 

 with that of a sphere of which the axis is AB, for the 



w 4 2*- 



attraction of that sphere r jtS ! X = "~ < 



attraction of the solid ADBH, (Fig. 1.) is, there- Plate 1, 



fore, to that of the sphere on the same axis as a 



to z- a, or as 6 to 5. 



To find the content of the solid ADBH, we need 

 only integrate the fluxionary expression for the content 



4"t 4 sr 



of the shell, viz. <r a. We have then a' = 



the content of the solid ADBH. Since the solidity 



of the sphere on the axis a is =^- s - tlie content of 



The 



