H9-] 



RECTILINEAR MOTION. 



59 



To integrate, put s=x?. The result will be different accord- 

 ing to the signs of /n, //, and v, which must be determined from 

 the nature of the particular problem. 



118. It is an empirical fact that the acceleration of bodies 

 falling in vacuo on the earth's surface is constant only for 

 distances from the surface that are very small in comparison, 

 with the radius of the earth. For larger distances the acceler- 

 ation is found inversely proportional to the square of the dis- 

 tance from the earth's centre. 



By a bold generalization Newton assumed this law to hold 

 generally between any two particles of matter ; and this as- 

 sumption has been verified by all subsequent observations. It 

 can therefore be regarded as a general law of nature that any 

 particle of matter produces in every other such particle, each 

 particle being regarded as concentrated at a point, an accelera- 

 tion inversely proportional to the square of the distance between 

 these points. This is known as Newton s law of universal grav- 

 itation, the acceleration being regarded as 

 caused by a force of attraction inherent in 

 each particle of matter. 



It is shown in the theory of attraction 

 that the attraction of a spherical mass, 

 such as the earth, on any particle outside 

 the sphere is the same as if the .mass of 

 the sphere were concentrated at its centre. 

 The acceleration produced by the earth on 

 any particle outside it is therefore inversely 

 proportional to the square of the distance 

 of the particle from the centre of the earth. 



o-- 



119. Let us now apply the general equa- 

 tions of Art. 117 to the particular case of Fl S- 30 - 

 a body falling from a great height towards the centre of the 

 earth, the resistance of the air being neglected. 



Let O be the centre of the earth (Fig. 30), /\ a point on its 



