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. 37
cubic corresponding to the critical quadratic specified above; one root of the cubic being
unity. But all the alternative possibilities for the general equation are not set out in
detail in the memoirs specified, so that all the possibilities for the limited cubic would
have to be considered independently.
Again, a considerable portion of Chapter v. of Konigsberger's treatise, already cited,
is devoted to the corresponding discussion for n equations; some difficulties as regards
adequacy of proof of the theorems, independently of the general statement, prevent me
from thinking the investigation entirely satisfactory, that is, if I understand it correctly *.
Some papers by Hornt may be consulted: further references will be found in them.
My intention in this paper is to take account of the different general cases that can
arise owing to the various possibilities of the form of the roots of the critical quadratic.
For this purpose, the method used by JordanJ for the corresponding discussion of a single
equation is adapted to the system of two equations. The different cases are:I. The quadratic has unequal roots:(a) neither root being a positive integer:
(b) one root being a positive integer, the other not:
(c) both roots being positive integers.
II. The quadratic has equal roots:(a) the (repeated) root not being a positive integer:
(b) the (repeated) root being a positive integer.
It should be added that a further assumption will be made for the present purpose,
viz. that the critical quadratic has not a zero-root. As a matter of fact, the existence
of a zero-root would imply (as for a single equation of the first order) that the reduced
form of the system belongs to a type different from that here considered.
* The investigation seems to imply (p. 397) that, taking are
n =2, the non-regular integrals of bB cB2
1'2 + x
1dY2 ' \ X=A A 2 + -- h B' 21 —h,'2 + B-s + fz2 + '"
x d=X2Y2+[x, Y, Y2]+ À 22
the unexpressed terms being terms of higher order in
when the real parts of X1 and XN are positive, arend, denoting z respectively. Th
' Z, t', 12, and l, ~'2 denoting z 1, z 2 respectively. The
Y_=- lx ic xl111+îvî+Àv4 12 only way in which z l can be a factor of x is by having
Y2=x a zc'x2+vî + 21v j22 B=0, and then za2 is not a factor of y; and similarly as
that is, x;À is a factor of Y, and xÀ2 a factor of Y2. But regards 22 and zl.
the integrals of t Crelle, t. cxvI (1896), pp. 265-306, ib., t. cxvII
(dx (1897), pp. 104-128, 254-266.
d = Xlx + azx + bzy + cy2 + Cours d'Analyse, t. III, ~~ 94-97.
z = + x + y + yy2