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. 63
When t=O and z = O, the term in this equation independent of T vanishes: but the
term in the first power of T does not vanish because q is not zero. Hence there is
one root, and only one root, of the cubic equation which vanishes when t= O and
z = 0; it is a regular function of t and zl in the immediate vicinity of t= O over a
region which is not infinitesimal. Actually solving the equation for this root, we find
MAI At A' \ I
T= A- - + - zl + higher powers of t and z1;
raGl or r1 ac
and then
P = A + -a + r + higher powers of t and zl.,7r1 \r ria,
Now knowing that such a solution of the dominant equation exists, we can obtain its
formal expression otherwise. Let
P = A' + Y, zlîmtnpr n
1T- 2x2 -zlint rn i
substitute these values in the dominant equations, expand their right-hand sides in the
form of regular series, and equate coefficients of zntn on the two sides. We find
rmn = K'<inn
Instead of actually evaluating K'mn, the analysis used to determine ymn can be adopted. To
this end, construct the value of y,,/, and, in its expression, effect the following changes in
succession:i. Replace every modulus of a sum by the sum of the moduli of its terns:
ii. Replace each denominator-factor I n + me, i and | n + (m + 1) b - 2 by r:
iii. Replace the coefficients of pl, rl, zP, tq, in gb and 0 by 1M m a/3n1 r lrlq, for all
values of mi, ni, pi, q1:
iv. Replace A I by A'.
The final expression, so modified, is K'mn. But the effect, upon the initial expression for
I Ymn, of each of these changes is to appreciate the value: hence, taking the cumulative
result, we have
I Ymn I < rmn.
Similarly
| mn I < rmn.
But the series
A' + EZ zlm tn Fmn
converges for a finite region round the origin t= O; hence the series
p = A + Z 7ymnzm tn )
= ZZXmnZimtn)