9. State and prove the rule for integration by substitution. Apply the substitution x = a cos2 + b sin2 0 to find the integrals √ − − √ (x − a)(b − x) √ √ (x − a)(b − x)dx. 10. Shew that the proper rational fraction p(x) and prove that it is equal to 4(b)(a) where (x) is any function which has p(x) for its differential coefficient. 12. Find a formula for the volume of a solid of revolution. revolves round the axis of x; find the volume generated. PURE MATHEMATICS.-PART III. The Board of Examiners. 1. Obtain a formula for the second differential coefficient of an implicit function of one variable. dzy dx2 2n-4 (n−2)a^—2xy { (n − 1)c2n-* + (n − 3)x^−2y"−2} ̧ (y"-1-a"-2x)3 2. Shew how to find the maximum and minimum values of p(x, y, z) where (x, y, z) = 0. Find the maximum and minimum values of x2 + y2 + ≈2 where 3. Find the polar equations of the tangent and normal at any point of a polar curve. Shew that three normals can be drawn from a given point to the curve r = a(1 + cos 0). 4. Trace the curves (i.) (x2-a2)y2 = x2(x2 + a2). (ii.) (0 + a)r = a(0 — a). 5. State and prove the rule for differentiating the integral b p(x, c) dx with respect to c. 7. Shew that by a proper choice of axes the equation of any two straight lines can be brought to the Through two straight lines in space two planes are drawn at right angles to one another; find the locus of their line of intersection. 8. Shew that the general equation of the second degree can always be reduced to one of the forms Ax2 + By2+ Cz2 = D, Ax2+By+2 Wz = 0, where one or more of the coefficients may be zero. Describe the form of the surface 9. Shew that there are two systems of generating lines on a hyperboloid of one sheet. Find the form of the equation to the surface when two of the generators are taken as two of the axes of coordinates. 10. Find the condition that the equation 0 Xd. + Tây + Z đã =0 may be derivable from a single primitive, and shew how to find that primitive when the condition is satisfied. Find the solution of yz(y2 + z2)dx + zx(z2 + x2)dy +xy(x2 + y2)dz=0. 11. Give the theory of the solution of the equations MIXED MATHEMATICS.-PART I. PASS AND FIRST HONOURS PAPER. The Board of Examiners. 1. Define the total force on a particle, and express a force of 1 pound weight in dynes. Two particles of masses m, m2 moving in the same straight line are connected by a straight light inelastic string. The forces on the particles are F, F, respectively in the direction of motion (exclusive of the tension of the string). Find the acceleration of the particles and the tension of the string. 2. A train is being drawn up an incline of 1 in 50 by an engine exerting a uniform pull of 100,000 poundals on the first carriage, and a dog running with the train gains on it 5 feet a second. The mass of the carriages is 60 tons, of the engine 40 tons, and friction is neglected. Find the accelerations of the train and dog and the work of the engine in one minute from starting. 3. Two spheres of equal mass having velocities v1 and vv, in the same straight line and in the same direction impinge. Assuming a coefficient of impact e shew that the velocities after impact |