93, 94] GRAVITATION. 97 



m is from m to m'. The last part of the statement is usually 

 expressed by saying that the force is an attraction. 



The Theory of Attractions is the theory by means of which the 

 resultants of forces exerted by the particles of gravitating bodies 

 are calculated. For our immediate purpose the most important 

 result of this theory is that, for a spherical body of uniform 

 density, (or having its mass so distributed that the surfaces of 

 equal density are concentric spheres,) the resultant attraction of 

 the body on a particle outside it is the same as that of a particle 

 at the centre, of mass equal to the mass of the body. This result 

 can be briefly (and less precisely) stated in the form that spheres 

 attract each other as if their masses were concentrated at their 

 centres. 



The direct application of the law of gravitation to celestial 

 mechanics requires for its complete statement the choice of a 

 frame of reference. We shall say at present that the origin is in 

 the Sun and that the lines of reference go out thence to stars 

 so distant that they have no observable annual parallax. (See 

 Chapter XIII.) 



Relatively to this frame the Earth and each of the planets 

 describe paths which are very approximately ellipses having one 

 focus in the Sun, and the radii vectores from the Sun to any 

 planet describe equal areas in equal times, while the square of the 

 periodic time in which any planet describes its ellipse is propor- 

 tional to the cube of the axis major of the ellipse. 



The statement just made is a statement of facts of observation. 

 Remembering the results of Articles 50 to 55, we can interpret it 

 by saying that each planet has an acceleration towards the Sun 

 varying inversely as the square of the distance. 



This statement is in accordance with the Law of Gravitation if 

 we suppose the mass of the Sun to be great compared with the 

 masses of the planets. 



The statement made in Article 91 concerning the field of force 

 near the Earth is also clearly included in the Law of Gravitation ; 

 for distances from the Earth's surface within which we can 

 manipulate bodies, the attraction of the Earth on a body is 

 practically constant. 



The constant 7 can be determined by direct experiment. 

 L. 7 



