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. 43
not, the process can be carried out m-1 times, but not more. Denoting the dependent
variables after all these transformations have been effected by u', v', we have equations in
the form
du'
t t= u + at+h (', v, t)
dv'
t -v = /v' + bt + k (u, ', t)
where /c, = - m +1, is not a positive integer; h, k are regular functions of their
arguments, vanishing with t and containing no terms of order less than 2. The
relation between the variables uz, v and u', v' is of the form
U = Pm-i + tm-U', V = Qn-i + tn-Vl,
where Pm_- and Qm-_ are algebraical polynomials of degree m -1 vanishing with t;
and u'= 0, v'= 0 when t= O. The coefficients a and b are algebraical functions of the
original coefficients.
The equations can possess regular integrals only if a is zero. For regular integrals
must be of the form
u =p pt + pt+..., v' = qt + qt2+...;
substituting these, remembering that h and k are then of the second order at least in
t, and equating coefficients of t in the first of the equations, we must have
Pi = pl + a,
which is possible for non-infinite values of pi only if a is zero.
Suppose now that a is zero. Since u' and v' (if they exist as regular functions
of t) vanish with t, we can assume
Zf'= tvl,, = t%2,
the sole transferred condition being that w7l and %2 are regular functions of t, which
now need not necessarily vanish with t. We have
t2 =h (tl1, tq2, t)=t2H (wl, 7; t),
t2 d= (, - 1) tr2 + bt + k (tW,, tq2, t)
t^^ t^t^+bt+2^,,t)
=(,c-l ) tw72 + bt + t2K (w1l, w)2, t),
where HI and K are regular functions of their arguments. The second equation shews
that, when t = O, then (c - 1), + b = 0; accordingly taking
b
72 = i- K+ 2
6- 2