Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

DIFFERENTIAL EQUATIONS. 71
In 8O,, let T be any one of the terms, and let T' be the value of T when o is
replaced by 1. As regards the numerator in T, it is the sum of a series of positive
quantities: and it is unaffected by the change of o, except that a10 is replaced by A,
that is, by a diminished quantity; hence the numerator of T' is less than that of T.
As regards the numerical denominator, each factor n + om- is replaced by n + m-1,
which is a greater quantity than the factor it replaces, unless m vanishes; but when
m = 0, then
- 2 - 7,
'n - o
n-1
because then n > 2. Also every factor n + om - K is replaced by n + m - c; the imaginary
portions (if any) of these two are the same, but the real part of the new factor is greater
than that of the old except when m =, and then they are the same. The number of
factors in the denominator is not greater than m + 2n- 1: hence
nI n + ~ crrm,- n + <m- 1 < (2 _ 7)f+n-i
n+ m — n+-m- c
< (2 - o)2m+2n
The changes made have diminished the numerator of T; thus
T' n + o-m -a- n + m -
T n+m -1 n +m-K/
< (2 - o)2m+2'
Remembering that 08m is a sum of terms T and bearing in mind the character of T,
we have
a nn <(2 _ ) j2+2.
Similarly
b'mn < (2 - ()2m+2n
Now the series
amn,,mtn, Z$bmn Omtn
converge absolutely for a finite region round the origin. Let this be defined by t tlr,
|0 s; and let -M1, M2 be the respective maximum values of the moduli of the series
within that region. Then
b <M2
amn < mrn bmn < smn;
and therefore
an < S fr M
(2 - a)2} (2- a)2
b'mn< I<-Is M2 n
(2-a)2} {(2-a)2}