OF THE ATTUACTIOIC OF GRAVITATING BODIES-. 



45 



produce a strain of '———, which is equal to the efiect of 

 half the weight of AC, actiaig at the distance AD. 



313. Theorem. In all cases of equili- 

 brium one general law prevails ; if motion 

 were imparted to the weights, their momenta 

 in the direction of gravity would be equal 

 and contrary. 



Taking the lever for an exampfe, it fs obvious that the 

 velochy of the bodies must be as their distances from the 

 fulcrum; and their weights being inversely in the same 

 ratio, their momenta must be equal, and always in direc- 

 tions perpendicular tt> the same line ; so that if the one 

 ascend vertically, the other must descend vertically. This 

 has been considered by some a* a sufficient foundation for 

 the demonstration of all cases of equilibrium, since it ap- 

 pears to be an absurdity to suppose, that any cause should so 

 act as to produce two equal effects, of which the one must 

 be contrary to the other, and to the operation of the com.- 

 mon cause. But it is more satisfactory to haye direct de- 

 monstrations in every case, and to deduce the general law 

 from all. 



Scholium. This.principle was extendetf still'further by 

 Jbhn Bernoulli, under the name of the law of virtual vela- 

 cities. Where the forces acting on the different bodies are 

 different, there is always an equilibrium when the sum of 

 all the products of the masses into the forces by which they 

 are actuated, and then into the initial velocities with 

 which they would be obliged to move, referred to the di- 

 rection of these forces, becomes equal to nothing. 



SECTION VIII. OF THE ATTHACTION OF 

 GRAVITATING BODIES. 



314. Definition. Graviuiting bodies 

 arc those of which the particles attract each 

 other with forces varying inversely as tlic 

 squares of the distances.. 



315. Theorem. All parallel sections of 

 a given cone or pyramid, supposed to be 

 gravit.iting surfaces, of a given evanescent 

 tliickness, attract a particle of gravitating 

 mutter placed at the vertex with equal force. 



The sections being considered as composed of evanescent 

 rectilinear figures terminated by the same right Unes, 

 meeting in the vertex, their areas are in the duplicate ratio 

 of their homologous sides, or of their distances from the 

 vertex (125, lai) ; and the whole areas, and the number of 

 material particles are in the same ratio ; therefore the in- 

 crease of the number exactly compensates for the increase 

 of the distance, and the forces acting in each line are the 

 same ; therefore the attractions of the whole section's are 

 the same. 



316. Theorem. A gravitating point 

 placed within a gravitatiiig spherical surface, 

 remains at rest. 



Conceive one half of the surface to be divided into eva- 

 nescent areolas, and cones or pyramids to stand on thcni 

 all, and to be continued through the given point as a ver- 

 tex, till they reach the surface on the opposite side : then 

 the inclination of each of two opposite cones to its base is 

 the same, and the magnitude of the section is the same as 

 if the sections were parallel, consequently the two opposite 

 and equal attractions destroy each other, and the same is 

 true of each particle of the surface, and of the whole sur- 

 face. 



317. Theorem. A gravitating point, 

 placed without a giavitating spherical sur- 

 face, or sphere, is attracted towards its cen- 

 tre with the same force as if the whole matter 

 of the surface or sphere were collected there. 



Call the radius unity, and the distance of the ordinate of 

 the sphere from the centre, x, then the fluxion of the curve, 



£, will be - 



, ,, ; (i4a), and if the ratio of the circum- 



^/^l— J^-fj 



ference of a circle to its diameter be that of p to 1, the cir- 

 cumference corresponding to a^ will be 'ip.y {\—xx), the 

 fluxion of the surface 2pi, and the superficial area itself 

 2//T, and, when xz^l, lp. The distance of the given point 

 from the centre being a, the absolute attraction of the 



2px' 



circular element of the superficies will be 



_ V'^' 



{a+xy + i — xj; 



-, and the effect in the direction of the axis 



oa-(-'2a.r4-l 



being diminished in the ratio of a+x to v^(aa-f 2ar-f-i), 



the fluxion of the attraction in that direction will be 



2p.(a-fr)..T /lp ax + 1 



a-t-2aa'-t-l)| \aa v ( 



-)■. 



and while x in- 



(aa-t-2aa'-t-l)| \aa ^ {aa-^-lax+l 



~— 2P 



creases from — 1 to ) , the fluent increases from — — to • 



