167-170] GRAVITATION. 151 



square of the distance therefrom. Thus for bodies near the Earth's 

 surface there is a correction of gravity for height above the Earth's 

 surface. In fact, if g is the value of the acceleration due to gravity 

 at the surface, and a the Earth's radius, the acceleration due to 

 gravity at a height h above the surface is 



Naturally, the correction only becomes sensible at heights 

 which can be reached with difficulty, as in the ascent of a balloon, 

 or at the top of a high mountain. 



The correction for depth below the Earth's surface, as for 

 instance at the bottom of a deep mine, depends on a result in the 

 Theory of Attractions, according to which the force exerted by a 

 uniform gravitating sphere at an internal point is proportional to 

 the distance from the centre. Thus, if h now denotes depth below 

 the Earth's surface, the acceleration due to gravity at depth h may 

 be taken to be 



g(a- h)/a. 



There are other corrections of gravity at least as important as 

 those here mentioned. One of them arises from the heterogeneity 

 of the material of the Earth, another from the fact that the Earth 

 is not spherical. Another correction depending on the choice of 

 the frame of reference will occupy us later in Chapter XIII. 



170. Examples. 



1. Consider the motion of a particle under the action of a uniform fixed 

 gravitating sphere, of density p and radius a, and suppose the particle to start 

 from rest at a distance b (>a) from the centre. It will move directly towards 

 the centre with an acceleration ^rrypa 3 fx 2 at distance x, so long as #>, and 

 when #=a, it will have a velocity given by 



Now suppose a fine tunnel is bored through the centre of the sphere in the 

 direction of motion of the particle. When the particle passes into the tunnel 

 its acceleration becomes ^nypx at distance x, and it moves with a simple 

 harmonic motion. The velocity at distance x is given by the equation 



^# 2 + -jf nyp x 2 = const., 



and the constant is determined from the expression given above for the velocity 

 at the instant of entering the tube. 



Prove that the velocity at the centre is 



[This is the result referred to by anticipation in Example 3 of Article 57. J 



