ATTRACTION. 



91 



Prop. V. 



Plate L. 

 Kg. 3. 



the solid ADBH is to that of the sphere on the same 



4 X , 5T , , . 4 1 



axis as a' to a> ; that is, as to -, or as 

 lo 6 la o 



8 to 5. 



It h?.s been supposed in the preceding investiga- 

 tion, that the particle on which the solid of greatest 

 attraction exerts its force is in contact with that solid. 

 Let it now be supposed, that the distance between 

 the solid and the particle is given; the solid being on 

 one side of a plane, and the particle at a given di- 

 stance from, the same plane on the opposite rule. 

 The mass of matter which is to compose the solid 

 being given, it is required to construct the solid. 



Let the particle to be attracted be at A ( Fig. 3. }, 

 from A draw A A' perpendicular to the given plane, 

 and let EF be any straight line in that plane, drawn 

 through the point A'; it is evident that the axis of 

 the solid required must be in AA' produced. Let 

 B be the vertex of the solid, then it will oe demon- 

 strated, as has been done above, that this solid is ge- 

 nerated by the revolution of the curve of equal at- 



. . . 4 1 



traction, (that of which the equation is^ ! ss a T x T 



x 1 ), about the axis of which one extremity is at A, 



and of which the length must be found from the 



quantity of matter in the solid. 



The solid required, then, is a segment of the solid 

 of greatest attraction, having B for its vertex, and a 

 circle, of which A' E or A' F is the radius, for its 

 base. 



To find the solid content of such a segment, CD 



4_ 2 



being = y, and AC = x, we have y* :r=a ! x T x*, 



4 1. 



mdiri/*x-=7ra' r x' T x 5rx 5 x = the cylinder, which 

 is the element of the solid segment. 



Therefore Jiry* x, or the solid segment intercept- 



r-4- xi 

 + 3 *' 



3 4 5 1 



ed between B and D must be ir a T iT * x i _i_ 



5 3 ' 



C. This must vanish when x = a, or when C comes 



to B, and therefore C.= -p- a 3 . The segment, 



therefore, intercepted between B and C, the line AC 



being x, is a' 



.4 s- 

 This also gives a', for the content of the whole 



K>lid, when x=0, the same value that was found by 

 another method at Prop. II. 



Now, if we suppose i to be r AA', and to be 

 given = b, the solid content of the segment becomes 



j-5 a * -J* - a y t> T + -q-bi, which must be made equal 



to the given solidity, which we shall suppose r= 70, 

 and from this -quation a, which is yet unknown, is 



to be determined. 

 3^ 

 S 



(&* 



If, then, for a T we put a,we have 



" 6t " 4 + f 43 ) = " ;3 ' or n u9 ~ j** 



* ="-> and u>- ( b { u = i?JL !^3 

 4-4* 12* 

 1 he simplest way of resolving this equation, would 

 ( be by the rule of false position. In some partfcular 



cases, it may he resolved more easily; thus, if ' " Ifsulkh. 



-%> 



0, 



9 f 

 i br u* 0, and ** = 



729 



-i'r.tha 



a=2-bT or a = b X (Y = &:/?* 

 4 V 4 / 64 



1. If it be required .to find l he equation to the p r0 p. VI. 

 superficies of the solid of greatest attraction, and also 

 to the sections" of it parallel to any plane passing 

 through the axis ; this can readily be done by help 

 of what has been demonstrated above. 



Let AHB (Fig. 4.) be a section of the solid, by a p LAre u 

 plane through AB its axis. Let G be any point in Fig. 4. 

 the superficies of the solid, GF a perpendicular from 

 G on the plane AHB, and FE a perpendicular from 

 F on the axis. Let AE = x, E F = 2, FG = v, then 

 x, 2, and v are the three co-ordinates by which the 

 superficies is to be defined. Let AB= a, EH = y, 



then, from the nature of the curve AHB, y- nm r ss r , 

 x\ But because the plane GEH is at right angles 

 to AB, G and H are in the circumference of a circle 

 of which E is the centre; so that GErEH= 

 Therefore EF' + FG'zrEH 1 , that is, z- + v*=y\ 

 and by substitution for y % in t,he former equation, 



z i + v'=aTx^x\ or (x* + z* + v*y = a* **, which 

 is the equation to the superficies of the solid of greatest 

 attraction. 



2. If we suppose EF, that is 2, to be given = h, 

 and the solid to be cut by a plane through FG and 

 CD, (CD being parallel to AB), making on the 

 surface of the solid the section DGC ; and if AK 

 be drawn at right angles to AB, meeting DC in K, 

 then we have, by writing b for 2 in either the pre- 



4 * 41 



ceding equations, b l -\-v z =:aJx r x% and v* = a r x T 

 x 1 b' for the equation of the curve DGC, the 

 co-ordinates being GF and FK, because FK is equal 

 to AE or x. 



This equation also belongs to a curve of equal at- 

 traction ; the plane in which that curve is being pa- 

 rallel to AB, the line in which the attraction is esti- 

 mated, and distant from it by the space b. 



Instead of reckoning the abscissa from K, it may 

 be made to begin at C. If AL or CK = h, then 

 the value of h is determined from the equation b*= 



a r h r h x , and if x = h + u, 11 being put for CF, 



v 2 = aT (h + u)T (h + u) 1 aJ h T + h\, or v' + 



(h + uy + b-- = a*(h+u)T, or (i> , + (/; + ) 2 + ^) 3 

 = a i (h + u)'. 



When b is equal to the maximum value of the or- 

 dinate EH, (Prop. III. 2.) the curve CGD goes 

 away into a point ; and if b be supposed greater than 

 this, the equation to the curve is impossible. 



The solid of greatest attraction may be found, and 

 it properties investigated, in the way that has now 

 been exemplified, whatever be the law of the attract- 

 ing force. It will be sufficient, in any case, to find 

 the equation of the generating curve, or the curve of 

 equal attraction. 



Thus, if the attraction which the particle C ( Fig. 1 . ) Fig. 1. 

 exerts on the given particle at A, be inversely as the 



