Mr. J. "W. L. Glaisher on a Property of Vulgar Fractions. 321 



a and 7. The measurement of S, however uncertain, renders 

 it probable that the spectrum of the negative pole in oxygen 

 forms part of the spectrum of lightning ; and on the whole I 

 should feel inclined to attribute the band « to oxygen. I 

 have shown in my paper on the spectrum of oxygen that this 

 spectrum of the negative pole is due to an allotropic modifi- 

 cation of oxygen (possibly ozone), and I have been able to 

 obtain it, though only temporarily, in the positive part of the 

 discharge. As regards the band y, I have some difficulty in 

 attributing it to nitrogen, and still think it probably due to 

 carbonic oxide. During the observations I certainly felt con- 

 vinced that it did not belong to the same spectrum as the chief 

 lines of nitrogen, and I made a note that, on the contrary, it 

 generally appeared together with /3. It seemed sometimes to 

 be present alone, and often to form the most prominent part 

 of the whole spectrum. As the lines of the capillary part of 

 an oxygen-tube are also present at the negative pole, together 

 with the bands distinctive of that pole, I can best express my 

 observations on the band spectrum of lightning by saying that 

 it resembles in a remarkable way the spectrum which is found 

 at the negative pole of a vacuum-tube filled with oxygen which 

 is slightly contaminated with carbonic oxide. 



LI. On a Property of Vulgar Fractions. 

 By J. W. L. Glaisher, M.A., F.B.S.* 



§ 1. ^ I iHB present paper relates principally to the following 

 A property of vulgar fractions : — If all the proper 

 fractions in their lowest terms having numerators and deno- 

 minators not exceeding a given number n, be written down in 

 order of magnitude, then each of these fractions is equal to 

 the fraction whose numerator and denominator are respectively 

 equal to the sum of the numerators and denominators of the 

 fractions on each side of it ; for example, if n — 7, the frac- 

 tions are 



and 



11112123143253456 



v v r a' v r 2' t v v r \ 



1 1+1 1 1+1 1 1+2 



' 6' 5' V f 3' 5' f 2' f V V f V 5' 6' 7' 



"> k = r. . ^ 



r, &c. 



6 7 + 5 y 5 6 + 4' 4 5 + 7 ; 



This property was enunciated by Mr. John Farey in the 



Philosophical Magazine for May 1816 (vol. xlvii. pp. 385-386), 



and was shortly afterwards proved by Cauchy. There is an- 



* Communicated by the Author. 



