92 



ATTRACTION. 



,tion 

 of Solids. 



' U i' 



Prop. VII. 



Plate L. 

 Vig. 5. 



Lemma. 



1 



m power of the distance, or as . _ , then the attrac- 



AE 



tion in the direction AE will be 



1 



AB 



AE=x, EC 



and if we 



make this = 



king 

 x 



1 



we have 



+ 



- = -,, or a' 



AC'"+*' 

 AE 1 



AC^= AF' r ma " 

 and AB s a, as before, 



xz={x 1 -\-y 1 ) , and x' + 



i m _J 



y 1 = a m+ x"-f", or ^ = a -t-i jj+ x\ 



If msfli or n-f- 1=2, this equation becomes /* = 



a x x 1 , being that of a circle of which the diameter 



is AB. If, therefore, the attracting force were in- 

 versely as the distance, the solid of greatest attrac- 

 tion would be a sphere. 



If the force be inversely as the cube of the distance, 



1 i 



ortw 3, and + l=4', the equation is y*=a lr x T x", 

 which belongs to a line of the 4th order. 



Si 



Ifra=4', and m+ 1=5, the equation is y'=za T x T 



x* ; which belongs to a line of the 10th order. 



In general, if m be an even number, the order of 

 the curve is m-J-1 >^2; but if m be an odd number, 

 it is m +- 1 simply. 



In considering the attraction of mountains in such 

 a manner as to make a due allowance for the hetero- 

 geneity of the mass, it is necessary to determine the 

 attraction of a half cylinder, or of any sector of a 

 cylinder, on a point situated in its axis, in a given 

 direction, at right angles to that axis. The solution 

 of this problem is much connected with the experi- 

 mental inquiries concerning the attraction of moun- 

 tains, and affords examples of maxima of the kind that 

 form the principal object of this paper. The fol- 

 lowing lemma is necessary to the solution. 



Let the quadrilateral DG ( Fig. 5. ) be the inde- 

 finitely small base of a column DH, which has every 

 where the same section, and is perpendicular to its 

 base DG. 



Let A be a point at a given distance from D, in 

 the plane DG ; it is required to find the force with 

 which the column DH attracts a particle at A, in 

 the direction AD. 



Let the distance AD = r, the angle DAE = <p, 

 DE (supposed variable) =y, and let EF be a section 

 of the solid parallel, and equal to the base DG ; and 

 let the area of DG = *. 



The element of the solid DF is m 1 y; and since DE, 



ancle DAE or <p, so that the element of the attrac- Attraction 



b ' m , . of Solids. 



tion of the column in the direction AD is <p cos. ^-"-"Y~ ' 



r 



,r m% /" "'* 



<p, and the attraction itself = J <p cos. <p ss sin. <p. 



When <p becomes equal to the whole angle sub- 

 tended by the column, the total attraction is equal to 

 the area of the base divided by the distance, and mul- 

 tiplied by the sine of the angle of elevation of the 

 column. 



If the angle of elevation be 30, the attraction of 

 the column is just half the attraction it would have, 

 supposing it extended to an infinite height. 



In this investigation, m' is supposed an infinitesi- 

 mal ; but if it be of a finite magnitude, provided it 

 be small, this theorem will afford a sufficient approxi- 

 mation to the attraction of the column, supposing 

 the distance AD to be measured from the centre of 

 gravity of the base, and the angle <p to be that which 

 is subtended by the axis of the column, or by its per- 

 pendicular height above the base. 



Let the semicircle CBG (Fig. 6.), having the Prop. VIII. 

 centre A, be the base of a half cylinder standing per- Plate L. 

 pendicular to the horizon, AB a line in the plane of Fig. C. 

 the base, bisecting the semicircle, and representing 

 the direction of the meridian ; it is required to find 

 the force with which the cylinder attracts a particle 

 at A, in the direction AB, supposing the radius of 

 the base and the altitude of the cylinder to be given. 



Let DF be an indefinitely small quadrilateral, con- 

 tained between two arches of circles described from 

 the centre A, and two radii drawn to A ; and let a 

 column stand on it of the same height with the half 

 cylinder, of which the base is the semicircle CBG. 

 Let z= the angle BAD, the azimuth of D; v = 

 the vertical angle subtended by the column on DF ; 

 a = the height of that column, or of the cylinder, 

 AD = x, AB, the radius of the base, ss r. 



By the last proposition, the column standing on 

 DF, exerts on A an attraction in the direction AD, 



which is = j-r-^- X sin. v. 



Now Drf=x, D/"=xz, and Drfx D/=o:zx. 

 Therefore the attraction in the direction AD is - 



ory=r tan.tp, =rtan.<p = r. -r, so that the ele- 



> J cos.ip 



P 



ment of the solid =m'r. -j- 



cos.p 



This quantity divided by AE 1 , that is, since AE : 



r z 



AD : : 1 : cos. , by ;, gives the element of the 



cos. q> 



attraction in the direction AE equal to 

 cos. <p z m' tp 



M X 



cos. tp 

 To reduce this to the direction 

 AD, it must be multiplied into the cosine of the 



x 



X sin. o = i! sin. v, and reduced to the direction AB, 

 it is x z sin. v X cos. z. 



This is the element of the attraction of the cylin- 

 dric shell orring, of which the radius is AD or x, and 

 the thickness x ; and therefore integrated on the sup- 

 position that z only is variable, and x and v constant, 



it gives x sin.D /"zcos.zrrxsin.DXsin.z for the at- 

 traction of the shell. When z=90, and sin.szrl, 

 we have the attraction of a quadrant of the shell = x 

 sin.u, and therefore that of the whole semicircle = 

 2x sin. v. 



Next, if x be made variable, and consequently v, 

 we have 2 /"xsin.u for the attraction of the semi- 

 cylinder. 



2 /"xsir 



