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.

III. On the Integrals of Systems of Differential Equations.
By Professor A. R. FORSYTH.
[Received, 28 July, 1899.]
INTRODUCTORY.
THE present paper deals with the character of the most general integral of a system
of two equations of the first order and the first degree in the derivatives of a couple
of dependent variables with regard to a single independent variable, the integrals being
determined with reference to assigned initial values. It will be seen that corresponding
results can be established for a system of n equations, of the first order and the first
degree in the derivatives of n dependent variables.
When the equations are given in the form
dy / \ dz
dx = f=y(dx = z)
then Cauchy's existence-theorem shews that, if x = a, y = b, z = c be an ordinary combination of values for the functions f and g, so that f and g are regular in the vicinity of
x = a, y =b, z= c, there exist integrals y and z of the equations, which are regular
functions of x and which acquire values b and c respectively when x = a; these solutions
are the only regular functions satisfying the assigned conditions; and it may be (but it
is not necessarily) the case that they are the only solutions of the equation (whether
regular or non-regular functions of x) determined by the assigned conditions.
If however a, b, c be not an ordinary combination of values, then the character of
the integrals of the equations depends upon the character of the functions f and g in
the vicinity. One important form, which includes a large number of cases, occurs when
a, b, c is an accidental singularity of the second kind for both f and g, that is, the two
functions are each of then expressible in the form
P(x - a, y-b, z - c)
Q( - a, y - b, z - c)'
where P and Q are regular functions of their arguments, each of them vanishing when
x = a, y = b, z = c. It is necessary to obtain an equivalent reduced form of the equations:
and one method is the appropriate generalisation of Briot and Bouquet's method as applied
to a single equation of the first order. This has been carried out in the case of
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