4. If a, B, 7, 8 be the roots of the equation x1 +8= O, prove that 5. The tangents at three points of a parabola make angles a, B. Y with the axis: prove that the area of the triangle formed by these tangents is to the area of the triangle formed by the normals as 1 (cota + cot B + cot y)2. 6. Prove that the radical axis of the circles of curvature at the extremities of any ordinate to a given diameter of a parabola passes through a fixed point. If the diameter be variable, the locus of this point is a parabola. 7. The normal to any curve at a point of inflexion is an asymptote of the evolute of the curve. 8. From a point P four normals are drawn to the semi-cubical parabola ay2 = 3, so that the sum of the angles which they make with the axis is constant (= a): prove that the locus of P is a straight line making an angle a with the axis, and passing through the focus of the curve. 9. A hypocycloid is generated by a circle of radius a rolling inside a circle of radius na, n being an integer. Determine the asymptotes of the curve which is the polar reciprocal of the hypocycloid with respect to the fixed circle. If the given curve have three cusps the equation of the reciprocal is 4x3 3 (x2 + y3) (x -3 a). Trace it. 10. A variable plane cuts off a given volume from a given ellipsoid, show that it envelopes one of three ellipsoids similar to the given ellipsoid. 11. Find the locus of the foci of the sections of the surface y1 + z2 = 4 ar made by planes passing through the origin. PURE MATHEMATICS, III. Examiner-MR. A. M. NASH, M. A. 1. Prove Taylor's theorem. State the necessary conditions and give an instance of its failure. 3. Eliminate the functions from the equation z= xf(y) + y$ (x). Determine the normal chord which cuts off the least area from 4. a parabola. 5. Find the equation of the circle of curvature at any point of the curve y = f (x). The evolute of the cardioid r = 2a (1 Cos 0) is another cardioid. 6. The circle of curvature at a point P of a parabola meets the curve again in Q: on PQ as diameter is described a circle cutting the parabola again in RR': prove that the envelope of RR' is another parabola. = 7. Trace the changes in the form of the curve xy2 + (x − a) (xb) (x-c) O as c changes from too. a, b being given positive quantities. 8. Show that the points of contact of the tangents which can be drawn from a given point to a curve of the nth degree lie upon a curve of the (n - 1)th degree. Hence prove that if the given curve have & double points and k cusps, the number of tangents which can be drawn from any point is n(n − 1) — 28 - 3k. 10. Determine the volume of an ellipsoid and the area of the surface of a sphere. 11. Define the pedal of a curve. The origins of pedals of a given area lie on a conic section and the conic has the same centre whatever be the given area. MIXED MATHEMATICS. Examiner-MR. W. BOOTH, B. A. 1. Explain in detail Foucault's method of determining the velocity of light. 2. Explain the formation of the primary and secondary rainbows, prove the formula for the radius of the primary bow 3. Prove that the general polar differential equation of the path of a ray of light through the earth's atmosphere the centre of the refraction and C is a constant, find the equation where 4. Find the geometrical focus of a pencil of rays after direct refraction through a sphere. 5. Prove the expression which is used for finding indices of refraction, and explain how the experiments may be made 6. Write an explanatory note on the “level error,” “collimation error," "deviation error" of a transit instrument. 7. Prove the formula lat = a — A cos h + A Sin 1" tan a Sin2 h, for determining the latitude by an observation of the pole star. 8. Prove Napier's analogies, or Gauss' Theorems. 9. Find the time occupied by the Sun in rising at a given place on a given day, (given apparent diameter). 10. Shew that the radius of curvature at any point on the terrestrial meridian supposing the earth an oblate spheroid is a- 2c3c sin2 7 where a and ac, are the axes equatorial and polar respectively, also that the distance of any point from centre = a c sin l. MIXED MATHEMATICS. Examiner-MR. W. BOOTH, B. A. 1 If O be the centre of mean position of a system of points A, B, C, etc. for a system of multiples a, b, c, etc. and P any arbitrary point then Σa A P2 = Σ a A O2 + Σ (a) O P2. 2. If three forces acting on a rigid body along three given right lines always have a single resultant, determine the locus of the line of action of this resultant. 3. Deduce the six equations of equilibrium of a rigid body from the principle of virtual velocities. When a system of forces is reduced to a pair of forces represented in magnitude and lines of action by two right lines, the volume of the tetrahedron formed by these lines is constant however the reduction is made. 4. A distribution of matter is made over a spherical surface whose density at any point varies inversely as the cube of its distance from a fixed point, show that the potential of the distribution at any point on the opposite side of the spherical surface is the same as that due to a certain quantity of matter at the given fixed point. 5. How does Newton solve the following; (a) The velocity at any point of an ellipse about a force in the focus is compounded of two μ не uniform velocities perpendicular to the radius vector, and perh pendicular to the major axis, (b) a body moves in an ellipse required the law of centripetal force tending to one of the foci, (c) a body moves in the circumference of a circle, required the law of centripetal force tending to any point in the plane of the circle. 6. Determine the general equations of equilibrium of a flexible inextensible string acted on by given forces and show from these equations that the osculating plane at any point passes through the applied force at the point. (a) A flexible uniform inextensible string whose two ends are fixed is acted on by a centre of force, if the string assumes the shape of an elliptic arc whose centre is centre of force, determine the law of force and the tension at any point. 7. A heavy uniform rough rod rests with its extremities on the interior of a rough vertical circle, determine the limiting position of equilibrium. MIXED MATHEMATICS. Examiner-MR. W. BOOTH, B. A. 1. A uniform circular plate attracts an external particle in its plane with a force varying inversely as the fifth power of the distance, shew that the resultant attraction = Mc where a is the radius and c is the distance of point from the centre. 2. A heavy uniform elastic ring is placed round a smooth vertical cone find how far it will descend. 3. If a heavy rough body rest upon two others which themselves rest upon a rough horizontal plane, shew that the three centres of gravity and the four points of contact lie in one plane. 4. Determine the condition of stable equilibrium for a floating body. 5. If the motion of a fluid in two dimensions be referred to polar co-ordinates r, e, and if u, v, denote the component velocities along and perpendicular to the radius vector find the component accelerations in the same directions. 6. If the depths of the angular points of a triangle below the surface of a fluid be a, b, c, shew that the depth of the centre of pressure below the centre of gravity is (a — b)2 + (b c) + (c 12 (a + b + c) = a)2 7. A ray of light passes through a medium bounded by parallel planes the density of which varies in such a manner that the index of refraction at any point 1+kx where x is the distance of the point from the plane on which the ray is first incident. The angle of incidence being a and the point of incidence the origin shew that the path of the ray is given by the equation 8. Rays emanate from the pole of a plane curve whose equation is f (r, p,) = 0 (1) shew that the equation of the catacaustic will be the result of eliminating and p between (1) and the equations 9. Find the effect of aberration on the right ascension and declination of a star. 10. When two planets are stationary with respect to each other 11. If the only impressed force be a central one velocity be that from infinity shew that the equation of the brachys tochrone is |