any "Farey Series" of ivhich the Limiting Number is given. 253 



function whose residue (in Cauchy's sense) is the generating 

 function to any given simple denunierant (see footnote to 

 American Journal of Mathematics, vol. v. p. 123) ; and I 

 became curious to know something about the value of Tj. I 

 had no difficulty in finding a functional equation which serves 

 to determine its limits (see Johns Hopkins University Circular, 

 Jan. and Feb. 1883). The most simple form of that equation 

 (omitted to be given in the Circular) is 



t/+t|+t|+t|+t| 



+...= 



r+j 



(where, when x is a fraction, Tx is to be understood to mean 

 Tj,j being the integer next below x); and from this it is not 

 difficult to deduce by strict demonstration that Tj/j 2 , when j 

 increases indefinitely, approximates indefinitely near to 3/7T 2 . 

 I have subsequently found that if ax be used to denote the 



sum of all the numbers inferior and prime to x, and U/= 2 ux, 



u i+2 ui +3u| +A\ji +...= & ,+i y +8 > 



J 2 3 4 3 



(where ILi',when x is a fraction, means the U of the integer next 

 inferior to x). From this equation it is also possible to prove that 

 TJj/j 3 , when j becomes indefinitely great, approximates to 1/tt 2 . 

 JJj, it may be well to notice, is the sum of all the numerators 

 of the fractions in a Farey series whose limit is^, just as Tj is 

 the number of these fractions. 



In the annexed Table the value of rx (the totient), of Tx 

 (the sum-totient), and of 3/7r 2 .^ 2 is calculated for all the 

 values of x from 1 to 500; and the remarkable fact is brought 

 to light that Tx is always greater than 3 / 7r 2 . x 2 (the number 

 opposite to it), and less than 3/7T 2 . (# + l) 2 , the number which 

 comes after the former one in the same table. 



I have calculated in my head the first few values of TLp, 

 and find (if I have made no mistake) that it obeys an analo- 

 gous law, viz. is always intermediate between 1 / 7r 2 . x z and 

 1/tt 2 .(^ + 1) 3 . 



It may also be noticed that when n is a prime number, 

 Tn is always nearer, and usually very much nearer, to the 

 superior than to the inferior limit — as might have been anti- 

 cipated from the circumstance that, when this is the case, in 

 passing from n— 1 to n the T receives an augmentation of 



n — 1, whereas its average augmentation is only — 2 (2n— 1). 



In like manner and for a similar reason, when n contains 

 several small factors Tn is nearer to the inferior than to the 

 superior limit. For instance, when n = 210, Tn = 13414 and 

 3/7f 2 .n 2 = 1340479. 



