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arbitrary constants arbitrary function auxiliary system Calculus of Variations Chap Chapter co-ordinates common integral complete primitive Crelle's Journal deduce derived determine dF dF differential coefficients dP dP dp dq dp dx dpi dpi dxj dq dp dq dy du du dx dx dx dy dy dx dz dxdy dxi dpi dpi dy _ dy dx dyj dp dz dx eliminate envelope species equa exact differential expression factor given differential equation given equation Hence homogeneous function identically satisfied independent variables infinite integrating factor involving Jacobi Lagrangean Last Multiplier linear partial differential memoir multiplied obtain ordinary differential equations partial differential equation particular integral Professor Boole proposition reduced relation represent respect result second member second order shew shewn singular solution substituting suppose system of ordinary system of partial theorem tion transformation vanish whence
Page ix - Researches on the Theory of Analytical Transformations, with a special application to the Reduction of the General Equation of the Second Order.
Page 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,...
Page 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,
Page 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.
Page 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...