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. 79
where K'm, J'mn are linear in the literal coefficients of p and r, and are integral
functions of km,,,' jm'n, such that m' Q m, n' ~ n, m' + n'< m + n. Also, from the form of
1i and 2,, K0'o = 0, J00 = 0; hence we have
koo = O.
But joo is undetermined, and it can therefore be taken arbitrarily: let its value be B,
where B is any arbitrary constant.
When the equations for kmn and j,,r are solved, in groups for the same value of
m + n and in succeeding groups for increasing values of m + n, they lead to results of
the form
kmn = Kmnn, Jmn = tmn,
where /Cm, tmn are sums of integral functions of the coefficients in -1 and 42, divided
by products of factors of the form n + mn.
The dominant functions are constructed as before. Let e denote the least value
of n +mel for integer values of m and n, so that e is a finite (non-vanishing) quantity;
and let 10l=, I C=C'. Also, let a common region of existence for the functions 4k
and.2 be given by the ranges \t ]r, \ r r., IPl h, rp T k; and within this region
let M be the greatest value of 1 11 and!%1. Then consider functions P and T, defined
by the equations
_______ P -
— E - IMn —Mr- 1-3r
E~-T cC + OP M- M i1/ (%M.
<Y \ - hr) t hr, k
Clearly
(e + ()P= e (T- C'),
that is,
p=- (T -C')
- -
The value of P is a root of a cubic equation which, when t = O and = 0, has
no term independent of P and has a non-vanishing term involving the first power of
P: so that it has one and only one root vanishing with t and u, and this root is a
regular function. To obtain its expression without actually solving the cubic, we take
P =:Kmnm,, 'M tn,
where K,, = 0: we expand the right-hand side of the dominant equations as a regular
function of t. ', P, T, and compare coefficients. The analysis that leads to the values
of Cmn, Lmn can be used to obtain the value of Kmn, by making appropriate changes
similar to those in the earlier corresponding case. These changes are now, as was the
case before, such as to make
| Kmn <K rmn, tmnnl <Krmn;