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.

36 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
n variables by Kônigsberger*, and in the case of two variables by Goursatt. For our
system, the most important reduced equivalent form is
t d =aU+teV+ yt+...=01(U, V,t)
/........................(A),
t = U+3, + t +.. = ( U V, t)j
u/t aU+/V~yt~O(UVt ~.(A),
where 0 and 0 are regular functions of their three arguments each of which vanishes
with U, V, t. The relations between the variables are
x-a=t', y-b=(b1+U) t, z-c=(c1+V)t4,
where 0, b,, are positive integers with no factor common to all three, and b1 and c,
are appropriately determined constants. The new conditions attaching to the dependent
variables U and V are that U = O and V = 0 when t = O; these correspond to the initial
conditions that y = b and z = c when x = a; and the matter to be discussed is the determination of integrals of the equations (A) subject to the condition that U= O and V= O
when t = O.
The integrals, so determined, are either regular or non-regular functions of t: their
existence and their character are affected by the nature of the roots of
(e- 01) (- - ~&)-221 = O,
which may be called the critical quadratic. Various theorems have been from time to
time enunciated in various investigations. Thus Picard+ proved that the equations possess
integrals, satisfying the required conditions and expressible as regular functions of t
provided neither root of the critical quadratic is a positive integer; and Goursat shewed~
that, if the real parts of each of the roots of the critical quadratic are negative, then
the equation possesses no integrals other than the regular functions of t satisfying the
required conditions. Also Poincaré|l and, following him, Bendixson~, have discussed the
integrals of n equations of the form
t d = 0r (ul, 2,..., Un), (r= 1,..., n),
the functions 0r being regular functions of their arguments and vanishing when ul = O,
2=O,..., u = 0: these can be made to include the system (A) by writing n= 3, and
taking the third equation in the form
du3
t dt = 3
with the condition that ua, u2, t13 all vanish with t. In this case, there is a critical
* Lehrbuch der Theorie der Differelitialgleichtungen, 743-745; see also his Cours d'Analyse, t. II, ch. i.
Leipzig (1889), pp. 352 et seq. ~ Amer. Joura. Math., vol. XI, p. 342.
t Amer. Journ. Math., vol. xi (1889), pp. 340, 341. Il Inaugural Dissertation, 1879.
+ Comptes Rendus, t. LXXXVII (1878), pp. 130-432, ~T Stockh. Ofver., t. LI (1894), pp. 141-151.