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.

44 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
we have r2 vanishing when t =. As regards r, there is, as yet, no restriction upon
its value when t= O; denoting it by A, we take
= A + gi,
where i vanishes when t =. Both, and r2 are regular functions of t. When the
values are substituted, A remains undetermined by the equations; and therefore an
arbitrary (finite) value can be assigned to A. The equations for r; and 2 now are
t d- tH / ~A+, t) A)f
t d = -1) 2+tK (A+, i + b tj
dt- =
with the condition that ~ and ~2 must be regular functions of t vanishing with t.
Let them, if they exist, be denoted by
= SE antn, 2= E bStn;
n=1 n=1
substituting in the equations which must be satisfied identically, and equating coefficients,
we have relations
nan =f/, (n- Kc + 1) bn = gz,
similar to those in the Case I (a).
These equations are treated in the same way as in the Case I (a). Since fc is
not a positive integer, no one of the coefficients of bn vanishes; and thence it is easy
to see that the whole of the treatment in I (a) subsequent to the corresponding stage
can, with only slight changes in the analysis, be applied to the present case. It leads
to the result that the power-series for ~i and 2 converge absolutely for a finite region
round t=O; and from the form of the equations for ~ and ~, it is clear that the
coefficients in the power-series will involve the arbitrary constant A.
Hence it follows that, unless the condition represented by a= O be satisfied, the
equations do not possess regular integrals vanishing with t = O. If that condition be
satisfied, the equations possess regular integrals vanishing with t =O and involving an
arbitrary constant: in other words, they possess a single infinitude of regular integrals
vanishing with t = O.
The condition represented by a = can be obtained from the original equations
du =mu+ot+0(u, y, t
td- = u + at + 0 (u, v, t)
pcii
as follows. Let
m - 1pt+
p=l
v= E cp tP + lt V;
p=l