II. On the integration of simultaneous differential equations

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Abstract
It is well known that a system of n —1 simultaneous differential equations of the first order connecting n variables always admits of n — 1 integrals, each of which is the form P = c , i. e. each of them is expressible by a function of the variables equated to an arbitrary constant. But when the number of the variables exceeds by more than by unity the number of the differential equations, no existing theory assigns the number of theoretically possible integrals, or guides us to their discovery. Yet cases such as this occur in problems of the greatest importance.

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