438 
MATHEMATICS: W. D. MACMILLAN 
Further investigations in this field have resulted in the three following 
theorems : 
Theorem I. If 7 is any positive number, rational or irrational, and 
if pn/n is a rational fraction such that \pn — ny\ ^ ^, and if 
1 " . 
An = ~ S {pK — Ky) is the arithmetic mean of the first n of the quan- 
tities {pK — Ky), signs considered, then the Hmit of A n, as 7t increases 
without limit, is zero. 
If y = p/q is rational with an even denominator q then for certain 
values of k there are two integers p^ which differ by unity such that 
\ pK — Ky\ = J; for one, the value is + J and for the other it is — |. It 
is supposed that such terms are taken alternately + | and — f . 
Theorem II. If 7 is a positive number, and if pnin is a rational 
1 " 
fraction such that \ pn — ny\'^\^ and if = - 2 | — «7 | is the 
n K=i 
arithmetic mean of the first n of the quantities \Pk — signs dis- 
carded, then the limit of An, SiS n increases without limit, is | if 7 is 
irrational or rational with an even denominator; but if 7 is rational with 
an odd denominator, 7 = p/q, then the limit of An is (q- — l)/4:q^. 
Theorem III. If 7 is a positive number, and if pn/n is a rational 
fraction such that \ pn — ny\-^^, and if Gn = ( ll\pK — 
the geometric mean of the first n of the quantities | — /C7 | , then 
the limit of Gn, as n increases without limit, is zero if 7 is rational, and 
is equal to l/{2e) where e = 2.71828 ... is the naperian base, if 7 is 
an irrational number which satisfies the condition 
+ i qn (qn + i) • . . (qn + s), 
where 7, expressed as a simple continued fraction, is 
7 = ^1 H , . y 
qn is the denominator of the principal convergent, and s any assigned 
positive integer independent of it. If 7 is an irrational number which 
does not satisfy this condition then Gn for large values of n oscillates 
between zero and l/(2e). 
1 
Ky I r is 
