for the Theory of Errors. 481 



of the series will all be correct to 4 places, and will have 

 errors ranging uniformly between + '000045. Of the first- 

 Fig. 1. 



named T T (7 of the series, half will have errors between + '000045 

 and +'000055, and half will have errors between —'000045 

 and —'000055 ; the distribution between these limits being 

 uniform*. The corresponding curve of error is given in fig. 2. 



Fig. 2. 



3. A similar case is the following. Suppose we have to 

 record successive positions of an index upon a fixed scale, 

 which is graduated in centimetres, and that readings are to 

 be taken to the nearest centimetre. If our judgment were 

 infinitely acute, the errors would lie uniformly between ±'5 

 centim. ; but in practice there will also be subjective errors, 

 the consideration of which is left to a later section (§ 10). 



4. Next let us consider the error introduced by friction 

 into the equilibrium position of a movable index. Suppose 

 that the index has one degree of freedom, and that if friction 

 were removed its vibrations would be simple harmonic ; the 

 frictional coefficient being the same at rest and at all speeds. 

 During a half-swing — say to the right — there will be a con- 

 stant force (or a constant moment) of friction urging the 

 index to the left, and its motion during the half-swing will be 

 harmonic and in the same half-period as if friction were 

 absent, the only difference being that the mean position of 

 the half-swing lies somewhat more to the left. In the return 

 half-swing there will be the same half-period, the mean 

 position being equally displaced to the right. The amplitude 



* That is, supposing that -00005 is added in the case of half of these 

 numbers, and subtracted in the case of the remaining half. 



