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. 41
To obtain the expansion of this root as a regular function, it is sufficient to determine
the coefficients in the power-series
X = Ait + A2t2 +... + Antn +...,
so that the equation
EX = -t t Xi+jtk
r pi+jrk
is identically satisfied; because the root which vanishes with t is the only root of the
cubic of that type, and the series for X is known to converge within the finite range
indicated. Clearly we have
Fn
where Fn is the coefficient of tn on the right-hand side of the equation for eX. When
this value of An is used for successive values of n, and the new expressions for
A1,..., An_ are substituted in Fn, the ultimate formal expression obtained for Fn is the
M
quotient of an integral algebraical expression in the coefficients i+ k by a power of e.
Pi+jrk
Comparing the quantities \fin and Fn, we note that a quantity greater than fnl is
obtained when in its numerator every term is replaced by its modulus; that this greater
quantity is further increased when the modulus of the coefficient of uivjtk in 01 or in
cp2 is replaced by i+j k; and that this increased quantity is still further appreciated
when every factor of the type i - 1 in the denominator is replaced by e. But, on
comparing the two coefficients an and An, it is clear that these three changes turn
fn into Fn; accordingly
fif, <E,.
Similarly for gn and Fn, so that
Ign < Fn.
Also |n - f | e - e31 > e;
hence ia n < An, ibn <A,,.
The series At + At2 + At3 +...
converges absolutely within a finite region round t=O; therefore also the series
a1t + at2 + a at3 +...,
blt + b2t2 + b3t8 +...,
converge absolutely within that region.
Hence the differential equations possess regular integrals which vanish with t. It is
not difficult to prove that they are the only regular integrals which vanish with t.
VOL. XVIII. 6