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.

64 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
converge absolutely: that is to say, the formal expressions p and r have significance, being
regular functions of zl and t. The equations accordingly have integrals
tl = pZl, t2 = T1,
of the characteristics indicated.
This completes the first stage of the proof.
11. For the second stage, let
ti = pzl + Ti, t = 7z + T2;
the equations for T1 and T2 are
dT1
dt = 1 T1 + 01 (pz, + T1, 71 + T2, t) -, (pzl, Tz1, t)
= Ti + T1 (T1, T2, zl, t)
dT2
tdt 2 22 + 2. (T1, T2, z1, t)
after substitution for p and r. Here 1f and f2 are regular functions of their arguments
vanishing when T1 = 0, T2 = 0; they contain no terns of aggregate dimensions lower than 2
in T1, T2, z1, t. In accordance with the statement in ~ 9, it has to be proved that these
equations possess solutions of the form
T1 = z2 e1, T2 = z02,
where ~1 and 20 are regular functions of t, zl, z2' it will appear that e2 =B (an arbitrary
constant) and 1-=0, when t=O. Substituting these values for T1 and T2, we find
t o + elzi al + e2z2a + (e2 - e1) e, =f (1, @2, z1, Z2, t))
at az, az,
a~2 a- CH)2 a0i 02 ^ / f
t + )z, + 2z =f2 (H1, OH zl, Z2) t)
the functions fi and f1 are regular in their arguments, every term involves (, or e2
or both, and a term involving,1 and 0, in the form 61A ~0, has also a factor z2^+-*.
If quantities e1, 20 exist, being regular functions of t, zi, z, and satisfying these
equations, the substitution of expressions of the form,1 = Y= Zplmn Z1lz2 mt, e, = S 2qln ZllZ2ntn,
in these equations must lead to identities. Accordingly, equating coefficients of zl1z2MXt
on the two sides of both equations, we have
{n + (I - 1) I + (m + 1) 2} Plmn = Ttlmnn,
(n + ^l + m^2) Çlmen = KImn,