72 SCIENTIFIC APPARATUS. 



In a simple harmonic motion it is found that the acceleration is 

 directed towards the centre, and is always proportional to the dis- 

 tance from it. In the case of elliptic motion it was proved by 

 Newton that the acceleration is directed towards the focus, and is 

 inversely proportional to the square of the distance from it. 



Let us now consider the circumstances under which these 

 motions take place. To produce a simple harmonic motion we 

 may take a piece of elastic string, whose length is equal to the 

 height of a smooth table; then fasten one end of the string to a 

 bullet and the other end to the floor, having passed it through a 

 hole in the table, so that the bullet just rests on the top of the 

 hole when the string is unstretched. If the bullet be now pulled 

 away from the hole so that the string is stretched, and then let 

 go, it will oscillate to and fro on either side of the hole with a 

 simple harmonic motion. The acceleration (or rate of change of 

 velocity) is here proportional to the distance from the hole ; that 

 is, to the amount of elongation of the string. It is directed towards 

 the hole ; that is, in the direction of this elongation. In the case 

 of the moon moving round the earth, the acceleration is directed 

 towards the earth, and is inversely proportional to the square of 

 the distance from the earth. 



In both these cases, then, the change of velocity depends upon 

 surrounding circumstances; but in the case of the bullet, this 

 circumstance is the strained condition of an adjoining body, 

 namely, the elastic string ; while in the case of the moon the cir- 

 cumstance is the position of a distant body, namely, the earth. 

 The motion of a projectile turns out to be only a special case of 

 the motion of the moon ; for the parabola which it describes may 

 be regarded as one end of a very long ellipse, whose other end 

 goes round the earth's centre. 



There is a remarkable difference between the two cases. The 

 swing of the bullet depends upon its size; a- large bullet will 

 oscillate more slowly than a small one. This leads us to modify 

 the rule. If a large bullet is equivalent to two small ones, then 



