Integration of homogeneous equations. Their general integral represents a system of similar curves. The equation (_a_ + _b x_ + _c y_) _dx_ + (_a’_ + _b’ x_ + _c’ y_) _dy_ = _c_, may be rendered homogeneous. Particular case where the method fails. How the integration may be effected in such case. Integration of the linear equation of the first order _dy_/_dx_ + P _y_ = Q, where P and Q denote functions of _x_. Examples. Remarks on the integration of equations of the first order which contain a higher power than the first of _dy_/_dx_. Case in which it may be resolved in respect of _dy_/_dx_. Case in which it may be resolved in respect of _x_ or _y_. Integrations of the equation _y_ = _x_ _dy_/_dx_ + φ(_d y_/_d x_). Its general integral represents a system of straight lines. A particular solution represents the envelop of this system. Solution of various problems in geometry which lead to differential equations of the first order. LESSONS 7-8. _Integration of Differential Equations of Orders superior to the First._ The general integral of an equation of the _m_ order contains _m_ arbitrary constants. (_The demonstration is made to depend on the consideration of infinitely small quantities._) Integration of the equation _d^{m}y_/_dx^{m}_ = φ(_x_.) Integration of the equation _d^{2}y_/_dx^{2}_ = φ(_y_, _dy_/_dx_). How this is reduced to an equation of the first order. Solution of various problems in geometry which conduct to differential equations of the second order. LESSONS 9-10. _On Linear Equations._ When a linear equation of the _m_^{th} order contains no term independent of the unknown function and its derivatives, the sum of any number whatever of particular integrals multiplied by arbitrary constants is also an integral. From this the conclusion is drawn that the general integral of this equation is deducible from the knowledge of _m_ particular integrals. Application to linear equations with constant co-efficients. Their integration is made to depend on the resolution of an algebraical equation. Case where this equation has imaginary roots. Case where it has equal roots. The general integral of a linear equation of any order, which contains a term independent of the function, may be reduced by the aid of quadratures to the integration of the same equation with this term omitted. LESSON 11. _Simultaneous Equations._ General considerations on the integration of simultaneous equations. It may be made to depend on the integrations of a single differential equation. Integration of a system of two simultaneous linear equations of the first order. LESSON 12. _Integrations of Equations by Series._ Development of the unknown function of the variable _x_ according to the powers of _x-a_. In certain cases only a particular integral is obtained. If the equation is linear, the general integral may be deduced from it by the variation of constants. LESSONS 13-16. _Partial Differential Equations._