A T T 



93 



A T W 



Now the angle v would have a for its sine if the 

 radius were Va z -\-x z , and so sin. v : 



BL 



wherefore the above expression is C . - = 2 a L 



(x + Va'+x 1 ) -f- C; and as this must vanish when 

 x=0, 2aL<z+C=0, and C = 2aLa, so that the 



fluent is 2 



flL *+W+: 



which, when .rrrr, 



gives the attraction of the semi-cylinder = 2aL 



; an expression very simple, and very con- 

 venient in calculation. It is probably needless to re- 

 mark, that the logarithms meant are the hyperbolic. 

 The limits of our work will not permit us to give 

 the other propositions which Mr Playfair has demon- 

 strated. We shall endeavour, however, to present 

 our readers with the results of his investigations. 



1. The attraction of the solid of greatest attrac- 

 tion is to the attraction of a sphere of equal bulk as 

 81 to 79. 



2. The cone which attracts a particle placed at its 

 vertex with the greatest force, is formed by the revo- 

 lution of a right angled triangle, the hypothenuse of 

 which makes an angle of 31 23' with the axis of ro- 

 tation ; or the cone of greatest attraction has the ra- 

 dius of its base nearly double of its altitude. The 

 attraction of a cone when a maximum, is about four- 

 fifths of the attraction of a sphere of equal solidity. 



3. A cylinder which exerts the greatest attraction 

 on a particle at the extremity of its axis, when the 

 radius of its base is to its altitude as 9 \/17 is to 

 8, or as 5 to 8 nearly. The attraction of a cylinder 

 of the preceding form is to that of a sphere of the 

 same solid content as 1218 to 1211.4. 



4. A semi-cylinder exerts the greatest attraction 

 upon a particle situated in the centre of its base, 

 when the altitude of the semi-cylinder is to the radius 

 of its base as 125 to 216. 



5. If the oblateness of a spheroid diminish, while 

 its quantity of matter remains the same, its attraction 

 will increase till its oblateness vanish, and the sphe- 

 roid become a sphere when the attraction at its 

 poles becomes a maximum. If the polar axis conti- 

 nue to increase, the spheroid becomes oblong, and the 

 attraction at the poles again diminishes. 



6. The force with which a particle of matter is 

 attracted by a parallelopiped, in a direction perpen- 

 dicular to any of its sides, may be determined by the 

 following rule. Multiply the sine of the greatest 

 elevation into the sine of the greatest azimuth of the 

 solid ; the arch, of which this is the sine, multiplied 

 into the thickness of the solid, is equal to its attrac- 

 tion in the direction of the perpendicular from the 

 point attracted. 



This rule will be understood from Fig. 7., where 



EM is the parallelopiped, having its thickness CE 



indefinitely small, A a particle situated without it, 



and AD a perpendicular to the plane CDMN. The 



greatest azimuth of the solid is the angle CAB, and 



BC 

 its sine is -^ . The greatest elevation of the solid 



(AD + DE)AN 



where n is the mea- 



is the angle BAL, and its sine is . , 



7. If a particle A gravitate to a rectangular plane, 

 or to a solid indefinitely thin, contained between two 

 parallel rectangular planes, its gravitation in the line 

 perpendicular to those planes will be equal to the 

 thickness of the solid multiplied into the area of the 

 spherical quadrilateral, subtended by either of those 

 planes at the centre A, or to the area of the spheri- 

 cal figure which the plane figure subtends at that 

 distance. 



8. An isosceles pyramid, with a square base, will 

 attract a particle at its vertex with the greatest force, 

 when the inclination of the opposite planes to one 

 another is an angle of 153. 



9. The force F which a parallelopiped BF exerts 

 upon a particle A, in a direction perpendicular to its 



a -a r , , m? t (AF+FN)AE 



sides, is r z=.i\a na'+BE. .Log v 



(AF+FM)AC 

 + BC.Log. ( l AD + DC j A M ; 



sure of the angular space, subtended at A by the 

 rectangle BD, and i, the angular space subtended by 

 the rectangle RF ; and AB=nr, and AK=o'. 



" This investigation," says Mr Playfair, " points 

 at the method of finding the figure which a fluid, 

 whether elastic or unelastic, would assume, if it sur- 

 rounded a cubical or prismatic body by which it was 

 attracted. It gives some hopes of being able to de- 

 termine generally the attraction of solids bounded by 

 any plane whatever ; so that it may some time or 

 other be of use in the theory of crystallisation, if 

 indeed that theory shall ever be placed on its true 

 basis, and founded not on an hypothesis purely geo- 

 metrical, or in some measure arbitrary, but on the 

 known principles of dynamics." The demonstration 

 of the preceding important results will be found in 

 the Transactions of the Royal Society of Edinburgh, 

 vol. vi. p. 187. (o) 



ATTRACTION and Repulsion of Floating 

 Bodies. See Floating Bodies. 



ATTRIBUTES. See Logic, God, and Theo- 

 logy. 



ATTRITION. See Friction and Mechanics. 



ATWOOD, George, F. R. S. a celebrated ma- 

 thematician, and natural philosopher, was born in 

 the year 1745. After receiving his education at 

 Westminster-school, he went to Cambridge, where he 

 was for some time a tutor, and afterwards a fellow of 

 Trinity college. The lectures on experimental phi- 

 losophy which he read to the university were much 

 admired ; and it was on this occasion that Mr At- 

 wood attracted the attention of Mr Pitt, who hap- 

 pened to be one of his auditors. When this states- 

 man came into power, he conferred a sinecure office 

 upon Mr Atwood in 1781, and employed him in all 

 his financial calculations. Mr Atwood invented a 

 very ingenious machine for exhibiting the phenomena 

 of accelerated and retarded motion, and for ascertain- 

 ing in a simple manner the quantity of matter moved, 

 the moving force, the space described, the time of 

 description, and the velocity acquired. It may be 

 employed also in estimating the velocities comrauni- 



Attraction 



Atwood. 



Fig. 8. 



