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admit appear application arbitrary constants assume auxiliary becomes Chap Chapter combination common complete primitive condition connecting consider consist construct contain corresponding deduce depend derived determine developed dF dF dF differential coefficients dP dP dx dy dy dy dy dz dz dz eliminate envelope equa equal equivalent established expression factor final function give given equation Hence identically idul important infinite integral involving known lead linear partial differential manuscript memoir method multiplied obtain ordinary differential equations original partial differential equation particular possible present Professor proposed proposition quantities reduced relation remains remarkable represent respect result roots satisfied shewn simultaneous singular solution substituting suppose surface theorem theory tion transformation values vanish variables whence
Side ix - Researches on the Theory of Analytical Transformations, with a special application to the Reduction of the General Equation of the Second Order.
Side 146 - ... that the solution of the two relevant systems ultimately depends on the solution of a system of ordinary differential equations of the first order, and that from these ordinary differential equations the given equation of the second order may be deduced independently of the assumption above mentioned. 1 shall also discuss the theory of the second integration. And I shall exemplify another method of solution connected by a remarkable law of reciprocity with the above method. First Investigation,...
Side 75 - 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,
Side 228 - T=c,, respectively, then we have Now v being determinable by an equation of the same form as u, it follows that of the above two values of u one must be assigned to v, so that the solution of the problem will be contained in the system or in the system The particular forms of the arbitrary functions <f, and ty will depend solely upon the nature of the problem under consideration.
Side 118 - Jacobi's method by finding an integral of the first partial differential equation, a process of derivation agreeing in principle with Jacobi's, only more extended, may lead us without further integration to a point at which the discovery of a common integral of the entire system will depend only upon the solution of a single differential equation of the first order susceptible of being made integrable by a factor. Failing this, it will enable us to convert the given system of partial differential...