376 NEWTON S PKINCIPIA. 



of observation may be tabulated in any form ; a good 

 way of exhibiting them is by deducing from them the 

 length of the seconds pendulum. If A be the length, we 

 have, 



9 = n 2 A ; 



thus g is always proportional to A. 



But if we wish to ensure accuracy in our results it is 

 clear that we must allow for the effects of all causes that 

 may affect the time of an oscillation besides gravity. These 

 corrections are called &quot; Reductions.&quot; Let us briefly con 

 sider what they are. 



1. If the centre of oscillation does not describe a cycloid, 

 allowance must be made for the alteration of time as de 

 pending on the arc described. This is called the &amp;lt;e reduc 

 tion to infinitely small arcs.&quot; If the arc of vibration be 

 n on each side of the vertical, then the time of an oscil 

 lation will be nearly 



VI 



hence the time of an oscillation must be divided by this 

 latter factor. If the arcs remained constant this would 

 be sufficient ; but it is found that the arcs continually de 

 crease by friction and resistance of air. Experiment 

 shows* that this decrease is in geometrical progression. 

 Taking this for granted, it is easy to deduce that the 

 mean time of one vibration will be 



where n, ri are the first and last arcs described, and m 

 their number. 



The time of an oscillation, as deduced from observation, 



* Borda. 



