KATE OF PENDULUM. 



of them be supposed to form pendulums, having the same angle 

 of oscillation. The arc of oscillation of the ball a will be a a"", 

 that of Swill be I W", that 

 of c, c c"", and so on. In 

 commencing to fall from 

 the points #, 5, c, J, e 

 towards the vertical line, 

 these five balls are equally 

 accelerated, inasmuch as 

 the circular arcs down 

 which they fall are all 

 equally inclined at this 

 point to the vertical line. 

 The same will be true if 

 we take them at any cor- 

 responding points, such as 

 a', b', c', d', e'. It may there- 

 fore be concluded, that 

 throughout the entire range 

 of oscillation of each of 

 these five pendulums, they 

 will be impelled by equal accelerating forces. 



Now it is shown by the principles of mechanics, that when bodies 

 are impelled by the same or equal accelerating forces, the spaces 

 through which they move are proportional to the squares of the 

 times of their motion ; therefore it follows, that the lengths of 

 these arcs of oscillation are proportional to the squares of the times. 

 But the lengths of these arcs are evidently in the same proportion 

 as the lengths of the pendulums, that is to say, the arc a a"" is 

 to 1 1"" as s a is to s b, and the arc b b"" is to c c"" as s b is 

 to s c, and so on. 



It follows, therefore, that the squares of the times of oscillation 

 of pendulums are as their lengths, or, what is the same, the times 

 of oscillation are as the square roots of their lengths. This 

 principle is easily verified experimentally. 



Let three small leaden balls be suspended vertically under each 

 other by means of loops of silken thread, as represented in fig. 3, 

 and in such a manner that they can all oscillate in the same 

 plane at right angles to the plane of the diagram, the suspending 

 loops not interfering with each other. 



Let the loops be so adjusted that the distance of the ball 1 

 below the line M N shall be 1 foot, the distance of the ball 4, 

 4 feet, and the distance of the ball 9, 9 feet. 



Let the ball 9 be put in a state of oscillation through small 

 arcs, and let the ball 4 be then drawn from its vertical position, 



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