176 Sir G. B. Airy on a Systematic Interruption in the 



In ranging the eye down the columns of differences; it is 

 seen at once that there are some differences much larger than 

 the rest. On close examination, it is seen that they belong 

 exclusively to the Vulgar Fractions whose denominators are 

 small — 1, 2, 3, &c. I now give the vulgar fractions and the 

 logarithms for several of these instances, accompanied with 

 the fractions next preceding and next following, to show the 

 magnitude of the differences. 



Denominator of Vulgar Fraction = 1. 



•29667 



i -30103 



2 -30544 

 49 



•00436 



•00441 



98 

 | 33 '47272 



' y * 47712 



100 1Q-MO 



33 48149 



•00440 



•00437 



99 



05 -59770 



'60206 



or -60656 



•00436 



■00450 



99 



20 

 b_ 

 1 



96 

 19 



100 

 67 

 3 



2 



69461 



69897 



70352 



•00436 



•00455 



95 



16 

 _6 

 1 



97 



10 



77360 



•77815 



•78265 



■00455 



•00450 



Denominator of Vulgar Fraction = 2. 



17393 



17609 



j* -17832 



65 



•00216 



•00223 



-§ -39794 

 I -40017 



00223 



00223 



37 -54177 



\ -54407 

 95 



27 



•00230 



•00229 



54636 



Denominator of Vulgar Fraction = 3. 



97 



73 -12345 



f -12494 



^ -12641 



•00149 



•00147 



98 



59 -22038 



4 -22185 



•00147 



•00149 



■22334 



43" -36653 



_7 

 3 



96 

 41 



•36798 

 •36949 



■00145 

 •00151 



The order of these numbers (which have been examined 

 much further) is sufficiently clear. For the critical or simple 



m 



values of — , the logarithmic differences immediately preceding 



. n -00440 

 or following these simple values are nearly. Now 



»00440 is nearly the logarithm of 1 + tttk- So that the values 



