When and where did corporate bodies first obtain charters? 7. Sketch the leading events of the Reign of Richard the Second. 8. Contrast his character with that of the Second Edward. Afternoon Paper. 1. What were the chief social and political consequence of the battle of Hastings? 2. Give the history of Robert Duke of Normandy as far as it is connected with that of England. 3. State the respective claims of Matilda and Stephen to the Crown of England and the termination of the contest. What was the Salique Law? Did it apply to England? 4. What were the Constitutions of Clarendon' and the Provisions of Oxford ?' How far did they become law? 5. Give the character of Richard the First. 6. From what period does the House of Commons date? Under what circumstances had it been previously convoked? 7. Give the account of the taking of Calais after the battle of Crecy. 8. What were the terms of the peace of Bretigni? Mathematics. FIRST CLASS. DIFFERENTIAL AND INTEGRAL CALCULUS. Morning Paper. α 1. Assuming the expansion for log. x in terms of x and its powers find the coefficient of h when x + h is written for x. tions of x; also find the nth differential coefficient of 3. Prove that f (b)-f (a) 1 b-a and def(x) have the same sign as long as a lies between a and b, provided the sign of de f(x) remains the same while changes continuously from a to b. 4. Expand f(x+h) in a series of ascending powers of h when x is indeterminate, and hence prove that ƒ (x) = (x) 1 (x = a)ƒ (x) 2 (x = a) + 5. Shew how to investigate the necessary condition that a given function may have a maximum or minimum when the requisite value of x makes it impossible to expand by Taylor's theorem. 6. Shew that curves which have a contact of the second order cut one another in the point of contact and generally that they cut or touch as the contact is of an even or odd order. 7. Investigate an expression for the radius of curvature to any point in a spiral in terms of SP and 0. Shew how to determine if the spiral has a point of inflexion. 8. Integrate 1 x √ x2 + a2 and a x + x2 9. Shew how to determine the partial fraction corresponding to a quadratic factor which has impossible roots and which occurs only once in the denominator. Hence integrate xm x+2 x2+x+1. x2 + 3 10. Shew how to integrate x + 1 where m and n are positive whole numbers. Afternoon Paper. 1. Given y=x. ε prove that =1+ + y 2. Let f (x. y) = o be an equation in which y is considered a function of x, write down the third derived equation. = 3. The equation to the evolute of an ellipse is (y b)3 + (x a)3 (a2—b2)3 where a and b are the semi axes of the ellipse, trace the evolute and shew to what portion of the ellipse each portion of the evolute belongs. 4. Trace the curves whose equations are y + x + 1 = (1−x)3 and 3+ y3 — a x3 = o 3+ 4x (1 + b. cos. x)2 also find the coefficient of a6 in the expansion of ε 6. Find the values of f. between the limits = o, x = ∞ and Sa (sin. x)" between the limits x = o, x = teger. π 2 n being a positive in 7. An egg in the form of a prolate* spheroid is placed in a cup of the same shape. The egg and the cup have equal minor axes and unequal major axes; find the volume in the cup not occupied by the egg. 8. If P be any point in a cycloid and O the corresponding position of the generating circle, shew that P O touches another cycloid of half the dimensions. OPTICS. Morning Paper. 1. Find the form and position of an image when the light from an object has been reflected at each of two plane mirrors. Within what space must the eye be situated to see a given point by reflection at a single plane mirror. 2. Find the focus of the reflected rays when a small diverging pencil is incident, directly on a convex spherical mirror, and shew that the conjugate foci always lie on the same side of the principal focus. 3. A rectangular box, at the bottom of which is a plane mirror, contains an unknown quantity of water; from the (given) angle at which a ray of light must enter through one of two small holes in the lid in order that after refraction and reflection it may emerge at the other, determine the height of water in the box. 4. In the three simple cases find expressions for the deviation of a ray of light in passing through a refracting prism in terms of the angle of the prism and the refractive index. 5. Divergent rays are incident upon a concave refracting surface find the position of the geometrical focus. When the conjugate foci are between the centre and the surface, find where the distance between them is a maximum. 6. In any ordinate perpendicular to the axis of a parabola a luminous point is situated, and a ray from this point parallel to the axis is incident upon the parabola, and after two reflexions again meets the ordinate, prove that the length of the path will be the same from whatever point in the ordinate the ray proceeds. * Generated by the revolution of an ellipse round its major axis. 7. Find the principal focal length of a convex meniscus the thickness of the lens being small but perceptible. 8. Explain the cause of short and long sightedness and find what form of lens is requisite to correct each defect. 9. If a and b be the distances of any two geometrical foci Q and q from the surface of a spherical refracting medium whose radius is r and Q', q' any other two geometrical foci shew that b 1 μα 1 a զզ b Q Q1 = r 10. By figure and explanation find the field of view in the Newtonian telescope. 11. Find the magnifying power of Galileo's telescope to an eye that can only see at the distance of 12 inches, having given the focal lengths of the object and eye glass to be 24 and 3 inches respectively. ASTRONOMY. 1. Shew that the earth's figure is that of a globe, differing little from a sphere; and describe the methods of finding its diameter, and also the diameter of its orbit. 2. Trace the changes in the position of the sun's diurnal circle as observed at a given place in the course of a year, and shew in what latitude he will be visible for 12 and for 24 hours respectively, when he has a given north declination. 3. Describe the vernier. The limb of the instrument is divided into spaces of 5′ each, and 9 divisions of the limb correspond to 8 of the vernier; to what accuracy will it read off? 4. Describe the transit instrument and the mode of making an observation with it. What corrections must be applied to an observation before we can ascertain the true position of the object? 5. Explain the cause of astronomical refraction, and shew how to find the coefficient of refraction by observations on two circumpolar stars. 6. Explain the method of finding the longitude of a place by measuring the moon's distance from the sun or from a bright star. 7. Explain the cause of aberration, and find its effect on the right ascension and declination of a star. 8. Find the time, magnitude and duration of a lunar eclipse. 1 9. The moon's diameter being 1867," and the moon's parallax 3421" find approximately in seconds the moon's apparent diameter at a place on the earth's surface having the moon in the zenith. 10. A circumpolar star is observed n times in 24 hours, all the intervals being equal. Shew that the sum of the cosines of the zenith distances = n sin. l. sin. d: where d, l are the star's declination and the latitude. 11. The zenith distances of a star, being observed on the same vertid-l cal circle, are z1, z, shew that tan. tan. = tan 2 d, l, being the star's declination and the latitude. 2 Z2 d + l cotan. 2 ; SECOND CLASS. CONICS AND HYDROSTATICS. Morning Paper. 1. Find the equation to a straight line drawn through a given point and perpendicular to a given straight line. Prove that the perpendiculars let fall from the angles of a triangle on the opposite sides pass through one point. 2. Find the polar equation to the circle, the pole S being without the circle. Deduce the length of the tangent to the circle from S, and find the limits of 0 the polar angle. 3. Whatever point in a parabola be taken for the origin of coordinates, its equation can be reduced to the form y2 = 4ax; where a is constant. 4. In the ellipse the rectangle contained by the perpendiculars from the foci upon the tangent at any point is equal to the square of the semi-axis minor. 5. If at the extremities of any two conjugate diameters to the hyperbola tangents be applied, so as to form a parallelogram, the area of the parallelogram is constant. 6. Define a level surface, and describe the level instrument used in surveying. 7. State the conditions under which a floating body will remain at rest, and determine the three cases of equilibrium by examining the position of the metacentre. 8. If a be the true weight of a body, w its weight in air, and w' its weight in water, then x = w + m (w-w') nearly: where m is the specific gravity of air. What is the arithmetical value of m? |