NEWTON S PRJNCIPIA. 375 



root of the length of the pendulum,, we see that short 

 pendulums must move much quicker than long ones. 

 This is an observation that every one must have made that 

 has entered a clock maker s shop. But the knowledge of 

 the above ratio enables us to correct the clock when the 

 length of its pendulum has been altered by temperature, 

 or, conversely, to alter the length when we wish to change 

 the rate. Suppose the pendulum to make n oscillations in 

 any given time M. Then clearly, 



M fg 



11 = v V r 



taking the logarithmic differential 



In _ L 8 



= &quot; 2 7&quot; 



This shows that a change of length equal to a fraction 

 of the whule length corresponds in any number of oscil 

 lations to a loss of a fraction of that number of oscil 

 lations. 



IV. Another use of the pendulum is to determine the 

 force of gravity and its variations over the surface of the 

 earth. When this has been done we have seen how the 

 true figure of the earth may be deduced. We have now 

 only to describe briefly how this application is made. 

 It will require, of course, great accuracy of observation. 

 The first requisite is to determine in the pendulum ex 

 perimented on the distance of the centre of oscillation 

 from the axis of suspension. This is the length called Z in 

 our formula. There is a variety of practical ways of de 

 termining this, which we shall not enter into now. The 

 second requisite is to determine the time of one oscilla 

 tion of this pendulum, and this is done by noting the time 

 of any large number of observations, and dividing by this 

 number ; thus any error made in observing the time of 

 beginning or ending is rendered insensible. The results 



BE 4 



