Definition of moments of inertia. How to calculate the moment of inertia of a body in relation to a straight line, when the moment in relation to a parallel straight line is known. How to represent the moments of inertia of a body relative to the straight lines which pass through a given point by means of the radii vectores of an ellipsoid. What is meant by the _principal axes of inertia_. Determination of the principal moments of inertia of certain homogeneous bodies, sphere, ellipsoid, prism, &c. LESSONS 43-45. _Calculus of Differences._ Calculation of differences of different orders of a function of one variable by means of values of the function corresponding to equidistant values of the variable. Expression for any one of the values of the function by means of the first, and its differences. Numerical applications; construction of tables representing a function whose differences beyond a certain order may be neglected. Application to the theory of interpolation. Formulæ for approximation by quadratures. Numerical exercises relative to the area of equilateral hyperbola or the calculation of a logarithm. LESSONS 46-48. _Revision._ General reflections on the subjects contained in the preceding course. ANALYSIS.--_SECOND YEAR._ CONTINUATION OF THE INTEGRAL CALCULUS. LESSONS 1-2. _Definite Integrals._ Differentiation of a definite integral with respect to a parameter in it, which is made to vary. Geometrical demonstration of the formula. Integration under the sign of integration. Application to the determination of certain definite integrals. Determination of the integrals ∫{(sin _ax_)/_x_}_dx_, and ∫{(cos _bx_ sin _ax_)/_x_}_dx_, between the limits _0_ and _x_. Remarkable discontinuity which these integrals present. Determination of ∫e^{-_x_^{2}}_dx_ and ∫e^{-_x_^{2}}cos _mx dx_ between the limits 0 and ∞. LESSON 3. _Integration of Differentials containing several Variables._ Condition that an expression of the form M _dx_ + N _dy_ in which M and N are given functions of _x_ and _y_ may be an exact differential of two independent variables _x_ and _y_. When this condition is satisfied, to find the function. Extension of this theory to the case of three variables. LESSONS 4-6. _Integration of Differential Equations of the First Order._ Differential equations of the first order with two variables. Problem in geometry to which these equations correspond. What is meant by their integral. This integral always exists, and its expression contains an arbitrary constant. Integration of the equation M _dx_ + N _dy_ = 0 when its first member is an exact differential. Whatever the functions M and N may be there always exists a factor _µ_, such that _µ_ (M _dx_ + N _dy_) is an exact differential.