is a ground product, (13431) not being a solution of the conditions.

Further

(12211) (12321) z = (12321) (13321),

wz = (13321),

(12211) (12321) y22 = (121) (25752),

(12211) (12321) ∞z3 = (13321) (25752);

so that there are no more ground products.

We have therefore in Case 6 obtained the new fundamental forms :

(12321),
(25752).

The investigation that has been given does not establish that the 13 forms obtained are ground products quâ the whole of the six cases, but it does prove that all the ground products are included amongst these 13. But it is clear that all forms in which a1 = 1 are necessarily ground products. This accounts for 9 of the 13 and it is easy by actual experiment to convince oneself that the remaining 4, viz.:

Hence the 13 ground products of order 5 are established.

Finally, to resume the foregoing, it has been shewn, in respect of the arithmetical function

n being any integer whatever, that all integral forms are expressible as products of

III. On the Integrals of Systems of Differential Equations.

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.

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 ƒ 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 and which acquire values b and c respectively when xa; 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 a) 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 ƒ 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 ƒ and g, that is, the two functions are each of them expressible in the form

P(x-a, y-b, z −c)
Q(x-a, yb, 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

n

variables by Königsberger*, and in the case of two variables by Goursatt. For our system, the most important reduced equivalent form is

where 01 and 0, are regular functions of their three arguments each of which vanishes with U, V, t. The relations between the variables are

where 0, 4, 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=0 and V=0 when t=0; 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=0 and V=0 when t=0.

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

(§ — a1) (§ — ß1) — α‚ß1 = 0,

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 and, following him, Bendixson, have discussed the

integrals of n equations of the form

dur dt

t · = 01 (U2, U2, ..., un), (r= 1,..., n),

(r=1,...,

the functions 0, being regular functions of their arguments and vanishing when u1 =0, U12 = 0,..., U2 = 0: these can be made to include the system (A) by writing n = 3, and taking the third equation in the form

with the condition that u1, us, us all vanish with t. In this case, there is a critical

* Lehrbuch der Theorie der Differentialgleichungen,

Leipzig (1889), pp. 352 et seq.

† Amer. Journ. Math., vol. x1 (1889), pp. 340, 341.

Comptes Rendus, t. LXXXVп (1878), pp. 430–432,

743-745; see also his Cours d'Analyse, t. III, ch. 1.

§ Amer. Journ. Math., vol. x1, p. 342.
Inaugural Dissertation, 1879.

¶ Stockh. Öfver., t. LI (1894), pp. 141–151.

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 Königsberger'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 Jordan‡ for the corresponding discussion of a single equation is adapted to the system of two equations. The different cases are:

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.

dx

the unexpressed terms being terms of higher order in 2, Š1, Š2, and Š1, Š2 denoting z11, z^2 respectively. The only way in which can be a factor of x is by having B=0, and then z is not a factor of y; and similarly as regards and 211.

+ Crelle, t. cxvi (1896), pp. 265-306, ib., t. cxvII (1897), pp. 104-128, 254–266.

+ Cours d'Analyse, t. III, §§ 94-97.