2. A function of x, y, z is transformed by the substitutions x = r sin @ cos, y = r sin 0 sin ø, z = r cos 0; 3. Prove the rule for integrating a definite integral with reference to a parameter independent of the limits, stating the conditions under which the rule holds good. has no meaning unless m and n are both positive. If m and n are both positive, prove that the integral is determinate and equal to T(m) г(n) 5. Find the area of the portion of the surface bounded by the cylinder x2 + y2 = a2, and the planes ≈ = 0 and = a. 6. Change the order of integration in the double integral 6. State some of the leading rules of the Brehon Law as to the alienation by tribesmen of their portion of the tribe land, comparing them with analogous ra es of Indian (Brahminical) Law. 7. What is meant by external sovereignty sovereignty, respectively? Mention briefly the chief questions whic reference to each kind of sovereignty. and internal arise with ld of law. 8. Name the five chief divisions of the (according to Holland), and give brief definition, or descriptions of the leading characteristics of each. 9. Give briefly Bentham's views as to an established church.' 10. How does Bentham treat the subject of duelling? What remedy does he suggest for the practice as it existed in his day? VI.—MATHEMATICAL SCIENCE. MATHEMATICS. [Full credit will be given for answering FCR-FIFTHS of this Paper.] SECTION A. L and DIFFERENTIAL AND INTEGRAL CALCULUS. PROFESSOR MOWEENEY. 1. State Lagrange's condition that a function of two independent variables should have a maximum or a minimum value; and discuss whether or not zero is a minimum of (y-x2) (y-2x2). Prove that x2y2 + ax2 + 2hxy + by3 has neither a maximum nor a minimum value if ab is negative, but that if a, h, and b are all positive, it has a minimum value zero, or - (h - √ ab)2, according as h2 < or > ab. 2. A function of x, y, z is transformed by the substi 3. Prove the rule for integrating a definite integral with reference to a parameter independent of the limits, stating the conditions under which the rule holds good. has no meaning unless m and n are both positive. If m and n are both positive, prove that the integral is determinate and equal to г(m) г(n). 5. Find the area of the portion of the surface bounded by the cylinder x2+ y2 = a2, and the planes ≈ = 0 and z = a. 6. Change the order of integration in the double integral SECTION B. SOLID GEOMETRY. PROFESSOR BROMWICH. 7. Show that the necessary and sufficient condition that the four planes l' = ny — mz, m' = lz - nx, n' = mx - ly, l'x + m'y + n'z = 0 may intersect in a line is Why do we find one condition in this special case, instead of four? Prove that the plane joining the line to the point (a, b, c) is given by the equation (l' + mc − nb) x + (m' + na − lc) y + (n' + lb − ma) z − (l'a+m'b+n'c) = 0. 8. Prove that, by a rotation through a small angle a about a line through the origin, the point (x, y, z) is displaced to (,, ), where ́=x+a(mz-ny), = x + a(mz - ny), n=y+a(nx-lz), (=z+a(ly-mx), and l, m, n are the direction-cosines of the axis of rotation. [N.B. a2 is to be neglected.] Explain how to distinguish between the positive and negative directions of rotation. What change is necessary if the axis of rotation does not pass through the origin, but is the line whose equations are given in question 7? 9. Show that the squares of the semiaxes of the section of the surface Prove further that the direction-cosines of an axis (2) are proportional to 1/(ar12+σ), m/(br12+σ), n/(cr ̧2 + σ). If n' is defined by ll' + mm' + nn' = 0, show that these conditions may be replaced by 11. Define a system of confocal quadrics, and prove that three such quadrics pass through a given point, cutting orthogonally at the point. Write down the equation of the enveloping cone to a quadric S, drawn from a point P, in terms of the parameters of the confocals to S which pass through P; and deduce (or prove otherwise) that two quadrics confocal to S will touch a given line, and that their tangent planes at the points of contact are perpendicular. 12. Show that, in the neighbourhood of an ordinary point on a surface, the equation to the surface may be put in the form 2% = ax2 + by2+ terms of higher order in x, y. Prove that the shortest distance between the normals at the origin and at (x, y, z) is, approximately, of length |(a - b) xy \/ (a2x2 + b2y2)3. Find where this shortest distance intersects the normal at the origin; and prove that the point of intersection lies within a length a-bab, whatever may be the position of (x, y, z). It is supposed that neither a nor b is zero; and the symbol X denotes the numerical value of the quantity X. |