ATTRACTION. 



89 



6f Solids. 



Attraction ne6ns matter must be formed, in order to attract a 

 particle given in position, with the greatest force 

 possible, in a given direction. 



Let A ( Fig. 1. Plate L. ) be the particle given in 

 position, AB the direction in which it is to be at- 

 tracted; and ACBH a section of the solid required, 

 by a plane passing through AB. 



Since the attraction of the solid is a maximum, by 

 hypothesis, any small variation in the figure of the 

 solid, provided the quantity of matter remain the 

 same, will not change the attraction in the direction 

 AB. If, therefore, a small portion of matter be 

 taken from any point C, in the superficies of the solid, 

 and placed at D, another point in the same superfi- 

 cies, there will be no variation produced in the force 

 which the solid exerts on the particle A, in the direc- 

 tion AB. 



The curve ACB, therefore, is the locus of all the 

 points in which a body being placed, will attract the 

 particle A in the direction AB, with the same force. 



This condition is sufficient to determine the nature 

 of the curve ABC. From C, any point in that curve, 

 draw CE perpendicular to AB ; then if a mass of 



matter placed at C be called r>0, . _ will be the at- 



A\* 



traction of that mass on A, in the direction AC, and 



IgJ y AE 



will be its attraction in the direction AB. 



Prop. II. 

 Fiff. I. 



AC 3 



As this is 



constant, it will be equal to , and 



AH 



therefore AB 2 X AE=AC 3 . 



All the sections of the required solid, therefore, 

 by planes passing through AB, have this property, 

 that AC 3 =AB S X AE ; and as this equation is suf- 

 ficient to determine the nature of the curve to which 

 it belongs, therefore all the sections of the solid, by 

 planes that pass through AB, are similar and equal 

 curves; and the solid of consequence may be con- 

 ceived to be generated by the revolution of ACB, 

 any one of these curves, about AB as an axis. 



The solid so generated may be called the solid of 

 greatest attraction; and the line ACB, the curve of 

 equal attraction. 



To find the equation between the co-ordinates of 

 ACB, the curve of equal attraction. 



From C (Fig. 1.) draw CE perpendicular to AB; 

 let AB=o, AE=x, EC=y. We have found AB 1 



XAE=AC 3 , that is, n'x=(x , +y) T , or a* x l = 

 {x*+y')*, which is an equation to a Line of the 6th 

 order. 



4 1 4 



To have_y in terms of x, x 1 -L-y'zza 7 x T , y'=a r 



* 'a/~ * 



x T x 1 , andy=x TV a T x T . 



Hence y=0, both when x=0, and when x=a. 

 Also if x be supposed greater than a, y is impossible. 

 No part of the curve, therefore, lies beyond B. 



The parts of the curve on opposite sides of the line 

 AB, are similar and equal, because the positive and 

 negative values of y are equal. There is also another 

 part of the curve on the side of A, opposite to B, 

 similar and equal to ACB ; for the values of y are 

 the same whether x be positive or negative. 



The curve may easily be conducted without ha- 

 ving recourse to the value of y just obtained. 



VOL. III. PART I, 



Let AB=a, (Fig. 1.) AC=s, and the angle BAC Attraction 

 = ?>. Then AE=ACxcos.<p=2Cos.ip, and so a* of Solids. 

 2 cos. <p=z 3 , or a 1 cos. <p=z 2 ; hence za\/ cos. <f. pTTteT^ 



From this formula a value of AC or z may be p; g . ] 

 found, if <p or the angle BAC be given ; and if it be 

 required to find z in numbers, it may be conveniently 

 calculated from this expression. A geometrical con- 

 struction may also be easily derived from it. For if 

 with the radius AB, a circle BFH be described from 

 the centre A ; if AC be produced to meet the cir- 

 cumference in F, and if FG be drawn at right angles 



AC Ap 

 to AB, then ~r^r= cos. <p, and so j=xv'Tnr= 

 Aii AH 



v / ABxAG=AC. 



Therefore, if from the centre A, with the distance 

 AB, a circle BFL be described, and if a circle be 

 also described on the diameter AB, as A KB, then 

 drawing any line AF from A, meeting the circle 

 BFH in F, and from F letting fall FG perpendicular 

 on AB, intersecting the semicircle A KB in K; if 

 AK be joined, and AC made equal to AK, the point 

 C is in the curve. 



For AK= y'AB X AG, fr om the na ture of the se- 

 micircle, and therefore AC=V / AB x AG, which has 

 been shewn to be a property of the curve. In this 

 way, any number of points of the curve may be de- 

 termined ; and the solid of greatest attraction will be 

 described, as already explained, by the revolution of 

 this curve about the axis AB. 



To find the area of the curve ACB. 



1. Let ACE, AFG (Fig. 2.) be two radii, inde- p;^ 

 finitely near to one another, meeting the curve ACB 

 in C and F, and the circle described with the radius 

 AB, in E and G. Let AC=z as before, the angle 

 BAC=<p, and AB=a. Then GE=a p, and the 

 area AGE = i 2 ip, and since AE* : AC 3 : : Sect. 

 AEG : Sect. ACF, the sector ACF=|z'<j>. But 

 z'rra* cos. <p, (Prop. II.) whence the sector ACF, or 

 the fluxion of the area ABCrr^a* q> cos. <p, and con. 

 sequently the area ABC=* a' sin. <p, to which no con- 

 stant quantity need be added, because it vanishes when 

 ip=:0, or when the area ABC vanishes. 



The whole area of the curve, therefore, is * a 1 , or 

 \ AB S ; for when <p is a right angle, sin. <p= 1 . Hence 

 the area of the curve on both sides of AB is equal to 

 the square of AB. 



2. The value of x when y is a maximum, is easily 

 found. For when y, and therefore y l is a maximum, 



Prop. HI. 



la? x T =2x, or 3x T =a T , that is x= =- 



3 T ^27 

 Hence, calling b the value of y when a maximum, 



2a 1 



b'=a-r x r = a' ( )= 



0-7S 97* 0"7T / 



27* 27' 



27^ 



4 



v/27 



, and so b 



t/27 



nearly. 



, and therefore a : b : : -^27: \/% or as 11 :7 



3. It is material to observe, that the radius of cur- 



4 1 



vature at A is infinite. For since y'zzaTxT x*, 



