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XX. On a Systematic Interruption in the Order of Numerical 

 Values of Vulgar Fractions, ivhen arranged in a Series of con- 

 secutive 'Magnitudes. By Sir G. B. Airy, K. C.B., Astro- 

 nomer Royal*. 



SOME years ago I prepared, with the aid of William Ellis, 

 Esq. (Assistant of the Koyal Observatory of Green- 

 wich), the numerical logarithmic values of all the Vulgar 



171 



Fractions — , in which m and n are prime to each other and 



n L 



do not exceed 100 ; and arranged them in a Table in order of 

 magnitude of the logarithms. The number of the fractions is 

 3043. On occasion of introducing this Table at a discussion 

 of the Institution of Civil Engineers, which involved the de- 

 termination of the number of teeth of wheels required to pro- 

 duce a given proportion of angular speed, the Council of that 

 Institution decided to print the Table in the Selected Papers of 

 their Transactions. 



The form of the Table may be understood from the following 

 specimen (taken at hazard): — 



Vulgar fraction. Logarithm. Difference. 



■ - 08190 -00009 



| -° 8199 -00020 



24 ' 08219 -00023 



m ■ -° 8242 -00011 



1 -° 8253 -00014 



m -° 8267 -00007 



I -° 8274 -00024 



23 -08298 



19 

 The average value of difference for the first 72 fractions is 

 •00010; for the following 116 fractions it is -00014; for the 

 next following groups of 116 each, it is '00016, '00016, 

 •00017, '00018, '00019, and so on, slowly increasing till we 

 come to high numbers. As representing the ordinary range 

 of differences, these numbers are too large, because they include 

 the effect of the special large differences, of which I have now 

 to treat. 



* Communicated by the Author. 

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