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Proceedings of the Royal Society of Edinburgh. [Sess. 
therefore no term of the series can be infinite. The series is a power-series 
in Jq 1 and Jq 2 , and it has been derived from another absolutely con- 
vergent power-series in Jq 1 and Jq 2 , namely, the series for H, by operations 
which are of an ordinary algebraical and trigonometrical combinatory 
character, except as regards the operation of introducing the divisors of 
the type (ms 1 + ns 2 ) (where m and n are positive or negative integers) in 
the integrations. We may therefore expect that the series will converge 
for sufficiently small values of Jq 1 and Jq 2 , unless the smallness of some 
of these divisors causes the series to diverge for all values of Jq 1 and Jq 2 , 
however small. Now the values of the integers m and n may indeed be 
so chosen that the divisor (ms x f ns 2 ) may be as small as we please: but 
| m | and I n | are then large, and since | m | and | n | are not greater than the 
order of the term, this small divisor can occur only in a term of high order, 
where it will be more or less neutralised by the high powers of Jq 1 and 
Jq 2 : and it was in fact shown many years ago by Bruns * that this state 
of things is consistent with the absolute convergence of a series. The 
example given by Bruns was the series 
21 
/y 1/1 sy 11 
■ 1 \ x2 
m — nA 
where q x and q 2 are proper fractions and A is a positive irrational number 
which is an algebraic number, i.e. a root of an irreducible algebraic equation 
A s + G x ~ 1 + G 2 A s _ 2 + . . . + G» = 0 
with integer coefficients G. If we multiply the numerator and denominator 
of any term in Bruns’ series by 
(m - n A') (m - n A") . . . 
where A', A", . . . are the other roots of the algebraic equation, then the 
denominator becomes a polynomial in m and n with integer coefficients : 
and as it is never zero, it must be at least equal to unity : while in the 
numerator we now have a polynomial in m and n of degree (s— 1) : whence 
it follows at once that Bruns’ series converges. 
The series (5) is much more complicated than Bruns’ series : and 
although the analogy so far as it goes is favourable to the convergence of 
(5), yet our opinion must rest mainly on the undoubted convergence of (5) 
in the case of particular systems where a test is possible. 
§ 6. Use of the integral found in § 4 in order to complete the integra- 
tion of the system. — Still restricting ourselves to Case I, in which the 
ratio sjs 2 is an irrational number, we now know two integrals of the 
dynamical system, namely, the integral of energy (which is obtained by 
* Astr. Nach. 109 (1884), p. 215. 
