capable of yielding, and the quantity of iron it includes in the form of pyrites? 8. A cadmiferous blende, on analysis, gave the following results How was the analysis made, and what is the rational formula deducible from it? 9. A mixture of malachite and azurite, upon analysis, gave— Oxide of copper, Carbonic acid, How was this analysis made, and what were the relative numbers of atoms of the green and of the blue carbonate in the mixture? 10. Chlorine, bromine, and iodine, have a similar action on a solution of caustic potash. What is it? 11. Enumerate the constituents of pig iron, and the mode of making its exact analysis. 12. Write the formulæ, the names, and the crystalline systems of the various native oxides of manganese, and explain Bunsen's mode of comparing their values as sources of chlorine. 13. Fluor spar and some other minerals are sometimes found as cubes with bevilled edges. What is the name, and what the notation, of the simple form to which such planes belong? 14. Why is the iodine of a solution of iodide of potassium only partially thrown down upon the addition to it of sulphate of copper; and how might the experiment be so conducted that the precipitation of the iodine would be complete ? 15. Write the notation of the planes which may replace, tangentially, certain terminal edges of an octahedron in the 5th system. 16. Give the formula and crystalline system of pyromorphite, and state how you would make its analysis when, as is usually the case, some of the phosphoric is replaced by arsenic acid. 17. An avoirdupois pound of magnesian limestone, first burned, and then boiled with bittern, left 4.5 ounces of magnesia; from such result deduce the composition of the dolomite, on the supposition of its including nothing but the carbonates of lime and magnesia. 18. Explain how you would analyze a mineral soluble in muriatic acid, and composed of phosphoric acid in combination with alumina, peroxide of iron, lime, and magnesia. 19. Give, in symbols, the composition of gypsum, heavy spar, and celestine; mention the crystalline system of each; and explain how you would analyze a mixture of such minerals. 20. Enumerate the metals comprehended in the three following groups: 1. Those whose solutions in the mineral acids are not precipitated by SH. 2. Those precipitated by SH, and re-dissolved by NH4 S. 3. Those precipitated by SH, but not re-dissolved by NH, S. FELLOWSHIP EXAMINATION. Examiners. RICHARD MAC DONNELL, D. D., Provost. HUMPHREY LLOYD, D. D. JAMES H. TODD, D. D., Regius Professor of Hebrew. JOHN TOLEKEN, M. D. CHARLES GRAVES, D. D., Professor of Mathematics. THOMAS STACK, M. A., Regius Professor of Greek. JOHN HEWITT JELLETT, M. A., Professor of Natural Philosophy. Mathematics, Pure and Applied. DR. HART. 1. What is the number of double tangents to a curve of the nth degree; and by what equation are their points of contacts determined? 2. This equation can be reduced, by means of a formula of Mr. Hesse, from the degree (n − 3) (n2 +2n − 4) to the degree (n− 2) (n − 3) (n+4). Apply the process to the case n = 4. 3. Explain the method of reducing the equation of the reciprocal of a curve of the fourth degree to the form S3 = 27T2; and show how to compute S and T? 4. How does Mr. Salmon obtain the equation of a system of right lines which pass through the points of inflexion of a given cubic? 5. Investigate the condition that a cubic should have a double point; and show that it can always be expressed in a concise form. 6. Given the four foci of a circular cubic; find the equation of the curve. 7. If a surface be generated by the motion of a horizontal right line which always cuts a given vertical right line and a given vertical circle; find the geodetic curvatures of an orthogonal trajectory of its genera trices. 8. Find the general conditions that one ruled surface may be capable of being developed on another; and compute the functions which enter into these conditions when one of the surfaces is an hyperboloid of revolution. 9. Find the general condition that two systems of lines may divide a curved surface into squares; and show that it is fulfilled by the lines of curvature of an ellipsoid. 10. The curvature of an ellipse varies in the triplicate ratio of the sine of the angle between the tangent and the focal radius vector. Prove the analogous property of lines of curvature of an ellipsoid. 11. If two systems of lines on a given surface cut one another at right angles, prove that the sum of the squares of their geodetic curvatures, together with the curvature of the surface, is equal to the sum of two differential coefficients of the geodetic curvatures. 12. Prove that if a curve of the fourth degree have two double points, the anharmonic functions of the systems of tangents drawn from these points are equal to one another. DR. GRAVES. 1. If the two particular integrals, u1 and 2, of the linear differential equation of the second order, D2y + A1 Dy + A2y = 0, are connected by the relation u2 = (u), being a function of known form; prove that the solution of the equation may be made to depend upon a quadrature. 2. If u1, u2, uз are the particular integrals of the linear differential equation, Show that this unsymmetrical triple integral can be transformed into a symmetrical group of single integrals. 3. Prove the following theorem :—In order that the equation Mdx+Ndy =0 may be made integrable by a factor μ, which is a homogeneous function of x and y of the nth degree, it is necessary, and it suffices, that on making y= vx, the function should assume the form ƒ (v). 4. Integrate the partial differential equation ps — qr = 0. 5. Explain the method of integrating the equation Day - Xy=0 by successive substitutions. Compare the result with that obtained by the method of separating symbols; and show that the series deduced is convergent. 6. Express the solution of the differential equation xDny + y = 0 in the form of a definite integral. 7. Supposing f(w) to remain finite within the limits of integration, and i to become equal to +∞, determine the limiting values of the definite integral corresponding to the cases where a and b have the same or contrary signs, or one of them becomes = o. 9. State and prove Dr. Boole's theorem for the expansion, in ascending powers, of o, off (π+p); π and p being distributive symbols which combine in subjection to the law of (r) u=Xƒ() pu, and X being a functional symbol which operates on in such a manner that f(π) =ƒ { $(π) }. 10. Apply Dr. Boole's method to sum the series 12. Show how the expression for Zux may be applied to approximate to the value of the product of the natural numbers from 1 to x; x being a very high number. 13. Determine y such a function of x that 14. The ordinates of a plane curve (B) are equal to the corresponding arcs of another plane curve (4), counted from a given abscissa a; what must be the nature of the curve (4) in order that the area of (B), taken between given abscissæ a and b, may be a maximum or minimum? 15. Prove that the plane curve which renders fo(p) ds, taken between two fixed points, a maximum or minimum, is such that all points of equal curvature are equally distant from the line which joins these extremities. 16. Discuss the surface whose quaternion equation is 17. Prove by means of the quaternion analysis that, if the vertex of a cone enveloping an ellipsoid be upon the focal hyperbola, the cone will be a right cone. |