Application of the Pendulum to Time-Keepers. 253 



whence 



and 



an 



Suppose x to be the augmentation or diminution of length in ques- 

 tion, and y the corresponding daily loss or gain in seconds, we 

 shall have 



an 2 an 2 an 2 a 



a' = a x = 



n' 2 (n^y) 2 n 2 ^p 2ny -\- y 2 j _ 2j/* 



<Z 

 f + L 



n 



nearly, neglecting as very small ; that is, ' 



nearly, from which we obtain, 



2 y a nx 



x = - , and y = - . 



Thus, if for any given rise of the thermometer, the pendulum 

 is lengthened one hundredth of an inch, we shall have for the 

 number of seconds lost per day, 



t = = 864QQ// x ' Q1 _ n // 



On the other hand, if a clock is known to keep time correct- 

 ly at a particular temperature, as 55 for instance, and at 32 is 

 found to gain 1" a day, we should be able to determine the cor- 

 responding diminution in length, or the contraction in the rod of 

 the pendulum, answering to this number of degrees ; thu s, 



0,006 inches. 



376. It will be seen, therefore, that by rendering the weight 

 of the pendulum moveable upon the red, and connecting it with a 

 micrometer screw, a correction may be applied for the expansion 



