252 Prof. Sylvester on the Number of Fractions contained in 



Philosophical Magazine, vol. xlvii. (1816) pp. 385, 386. In 

 1879 Mr. Glaisher published in the Philosophical Magazine 

 (pp. 321-336) a paper on the same subject containing a proof 

 of their known properties, an important extension of the sub- 

 ject to series in which the numerators and denominators are 

 subject to distinct limits, and a bibliography of Mr. Goodwyn's 

 tables of such series. Finally, in 1881 Sir George Airy con- 

 tributed a paper also to the Philosophical Magazine of that 

 year, in which he refers to a table calculated by him " some 

 years ago," and printed in the Selected Papers of the Transac- 

 tions of the Institution of Civil Engineers, which is in fact a 

 Farey table with the logarithms of the fractions appended to 

 each of them. Previous tables had only given the decimal 

 values of such fractions. The drift of this paper is to point 

 out a caution which is necessary to observe in the use of such 

 tables, and which limits their practical utility : this arises 

 from the fact of the differences receiving a very large aug- 

 mentation in the immediate neighbourhood of the fractions 

 which are a small aliquot part of unity — a fact which may be 

 inferred a priori from the well-known law discovered by Farey 

 applicable to those differences, but to which the author of the 

 paper makes no allusion. 



In addition to the tables of Farey series by Goodwyn, 

 Wucherer, an anonymous author mentioned in the Babbage 

 Catalogue, and Gauss, referred to by Mr. Glaisher in his 

 Report to the Bradford Meeting of the British Association 

 (1873), may be mentioned one contained in Herzer's Tabellen 

 (Basle, 1864) with the limit 57, and another in Hrabak's 

 Tabellen- Werk (Leipsic, 1876), in which the limit is taken at 50. 



The writers on the theory are: — Cauchy (as mentioned by 

 Mr. Glaisher), who inserted a communication relating to it in 

 the Bulletin des Sciences par la Socie / te / Philomathiquede Paris, 

 republished in his Exercices de Mathdmatiques ; Mr. Glaisher 

 himself (loc. cit.); M. Halphen, in a recent volume of the Pro- 

 ceedings of the Mathematical Society of France; and M. Lucas, 

 in the next following volume of the same collection. I am 

 indebted to my friend and associate Dr. Story for these later 

 references. 



For theoretical purposes it is desirable to count -\ as one 

 of the fractions in a Farey series. The number of such frac- 

 tions for the limit j then becomes identical with the sum of 

 the totients of all the natural numbers up to^' inclusive — a 

 totient to x (which I always denote by rx) meaning the num- 

 ber of numbers less than x and prime to it. Such sum, i. e. 



2 rx, I denote by Tj. My attention was called to the sub- 

 ject by this number Tj expressing the number of terms in a 



