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.

54 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
NON-REGULAR INTEGRALS.
6. It has been seen that, either in general or subject to certain conditions, the
equations
du
tddt = al U + l,1 +C t Y t..*= 01 ( U V, z)
t d= aU +/3,V+yt +... = 0(U, V, z)
possess regular integrals which vanish with t: and these are unique as regular integrals.
Denoting them by ul, v1, let
U=u + x, V=v + y;
so that if functions x and y exist, different from constant zero, they are non-regular
functions of t, and they must vanish with t because U, u1, V, v1 all vanish with t. Then
dx
t = (ul + x, v1 +y, t)- 01 (ul, v, t)
=! Â^y aw ^(l,- ) V,Iï n(iau,~ av) y
t n i (X +y a) 02 (1, VI, t)
are equations to determine x and y. On the right-hand sides there are no terms
involving t alone; the only terms of the first order are alx + /y, a2x + 2y respectively;
and the coefficients of the other powers of x and y are functions of t and of u1, v1,
that is, after substitution of the values of u1, vy, these coefficients are regular functions
of t. Hence we may take the equations in the form
tdt = alx + /31y + à (x, Y, t)
dy
t=dYa2x-4- /32y + s1(xy, t) )
where àI, and 42 are regular functions of x, y, t, vanishing when x = 0, y = 0, and containing no terms of dimensions lower than 2 in x, y, and t. The dependent variables
x and y, if they exist as other than zero constants (which manifestly satisfy the
equations), are to be non-regular functions of t which vanish when t = 0.
It is convenient to transform the equations by linear changes of the dependent
variables, as was done for the discussion of regular integrals: the new forms depending
upon the roots of the critical quadratic
(F- a) ^ - Pz)- aAP1