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It is a fundamental proposition of analysis that a system ofn differential equations of the first order containingn + 1 variables admits ofn integrals, each of which is expressed by a function of the variables equated to an arbitrary constant. But when a system ofn differential equations of the first order connectsn +r variables,r being greater than unity, no existing theory assigns in a general manner the number of theoretically possible integrals of the above .species, or shows us how to discover them. Yet such cases are of great importance.

philosophical transactions of the royal society of london

XXX. On simultaneous differential equations of the first order in which the number of the variables exceeds by more than one the number of the equations