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. 73
13. The main theorems as to the equations
t - t + at+fi (t, t2, t)
t dt2 Kt2 + bt +tf2 (tl, t2, t))
dt
so far as concerns the non-regular solutions, are:When a is not zero, so that the equations do not possess any regular solutions that
vanish with t, they possess non-regular solutions that vanish with t. If Kc have its real
part positive, not itself being a positive integer, there is a double infinitude of such solutions;
they are regular functions of t, tK and t log t. If K have its real part negative, there is
only a single infinitude of such solutions, they are regular functions of t and t log t.
When a is zero, so that the equations possess a single infinitude of regular solutions
vanishing with t, then if Kc have its real part positive, not itself being a positive integer,
there is a single infinitude of non-regular solutions vanishing with t which are regular
functions of t and tK; but if K have its real part negative, the equations possess no nonregular solutions vanishing with t.
These theorems can be established by analysis and a course of argument similar
to those which have been adopted, wholly or partially, in preceding cases. The actual
expressions for the integrals, when a is not zero, are
t, = aO + At + Sgln,,n lOmt'
t2 = - t +B+ s t+ mn y Omt tn
where the summation is for values of 1, qn, n such that 1 + m + n > 2, the coefficients
A and B are arbitrary, [ denotes tK and 0 denotes t log t.
When a is zero, all the coefficients gmn, hlmn for values of m > 0 vanish; so that
0 disappears from the expressions for t1 and t2. The resulting expressions then can be
resolved each into the sum of two functions: one a regular function of t which
involves A, the other a regular function of t and ' which involves B, and vanishes
when B = 0.
It may be noted that a slight degeneration occurs in the solutions when Kc is the
reciprocal of a positive integer; a regular function of t and tK is then merely a
regular function of tK.
When the equations in their first transformed expression are
du
t d = mu + at + 0 (u, v. t)
dt
t = v + /3t + (u, V, t))
the general results are the same as above; the value of KC is -m+ 1, and the
critical condition, which is represented by a =0, is stated at the end of ~ 2.
VOL. XVIII. 10