FOURTH CLASS. EUCLID AND ALGEBRA. Appendix C. Morning Paper. (1.) In any right angled triangle, the square which is described upon the side subtending the right angle is equal to the squares described upon the other two sides which contain the right angle. Is this proposition included in any more general one? (2.) To divide a given straight line into two parts, so that the rectangle contained by the whole and one of the parts, shall be equal to the square of the other part. Can this be solved arithmetically? If so, find approximately into how many parts the given line must be divided. (3.) Prove that the opposite angles of any quadrilateral figure which can be inscribed in a circle are together equal to two right angles. (4.) If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides or those produced proportionally, and if the sides or the sides produced be cut proportionally, the straight line which joins the point of section shall be parallel to the remaining sides of the triangle. Hence shew how a line may be drawn on the ground through a given point, parallel to a given straight line by means of a piece of string. (5.) Every solid angle is contained by plane angles, which together are less than four right angles. (6.) A person who had a 9 anna share in an Indigo factory, made his younger brother a present of 75 per cent. of his share, and sold the remainder to his cousin, who soon after purchased of the younger brother's share, but now offers to dispose of half his interest in the factory for Rs. 7,000. Estimating at the same rate, what was the value of the whole factory, and each brother's share? (7.) If a and b be two integral numbers prime to one another and the product a + c be divisible by b, show that c must be divisible by b. Find the form of the denominator of a vulgar fraction in its lowest terms when it is reducible to a terminating decimal. Isso reducible? (8.) To extract when possible the cube root of a binominal surd, one of whose terms is a rational quantity, and the other a quadratic surd. In equation (3) explain the result when the values of x and y assume the form of g (10.) Insert m harmonical means between a and b. The distance between Calcutta and Barrackpore is 14 miles; now if a single stone were laid upon every yard of that distance, and the first one was a yard from the basket, what distance would a man travel in bringing the stones one by one to the basket? (11.) Write down the number of variations of m things taken r and r together. Find the greatest term in the expansion of (1+x) without regard to sign m being positive and x a proper fraction. Will the same investigation hold when m is negative? (12.) Find the amount of an annuity, left unpaid for m years, at simple interest. Explain why it is not consistent with the principle of simple interest to consider the amount of an annuity to be sum of the present values due at the periods 1, 2, 3, years. .... m, (13.) Investigate a rule for forming the consecutive converging fractions. How may converging fractions be employed to find the logarithm corresponding to any number? (2.) Within a given parabola inscribe the greatest parabola, the vertex of the latter being at the bisection of the base of the former. (3.) Investigate a differential expression for the radius of curvature, and shew that it is identical with Newton's expression, с In the curve y=() the ordinate at any point is a mean proportional to the (4.) Define the evolute of a curve. Investigate the property on which it depends; find the evolute to the cycloid. (5.) Determine the nature of the curve whose equation is y3+x3-ax2 = 0, find the maximum ordinate, and point of inflexion. Trace and find the area of the curve whose equation is x1 + y1 — a2xy = 0 (6.) If in the radius vector SP of a parabola, (the vertex of which is A, and Sy the perpendicular from the focus S upon a tangent at P) a point Q be taken, such that SA : Sy=SQ: SP, find the equation to the curve which is the locus of Q; trace the curve, and show that the areas of the curve and parabola between the vertex and the latus rectum of the parabola are as 3: 4. = (7.) Shew how to find the length of a curve referred, (1) to rectangular co-ordinates, (2) to polar co-ordinates. Prove that the length of the curve whose equation is + y3 intercepted between the axes of r and y is За a3 (8.) Find the volume of the solid generated by the revolution, abeut the axis of x, of the lemniscata the equation of which is (1.) State the steps in the reasoning by which it is shewn that f (x + h) admits of development in a series proceeding by ascending positive and integral powers of h. dy (3.) Define a multiple point, and show from the definition that if be obtained from dx the equation to the curve made free of radicals, the co-ordinates of the multiple point will make it assume the form 0 0 Take as an example the curve ay2 = a2 (x2 — y2), and determine the direction of its branches at the multiple point. (4.) A curve is convex or concave to the axis of x, according as same sign as the ordinate. d'y has, or has not, the Determine Appendix C. Determine the minimum value of (x-a)" m being odd. (5.) Find the differential expression for the radius of curvature, and shew that it agrees with Newton's. If y and x be functions of a third variable 0, the expression for the radius of curva determine what this expression becomes when is the arc of the curve. (6.) Trace the curves defined by the equations (7.) Investigate the differential expression for a surface of revolution; and find the surface generated by the revolution of the lemniscata, the polar equation of which is 2 a2 cos 20. = (8.) Find the locus of the intersection of the perpendicular, drawn from the vertex, and tangent to any point of a parabola. Trace the curve and find the area between the curve and its asymptote. (1.) Shew how to transform an equation into one which shall want the second or third term; under what circumstances can both be made to disappear at one operation? Form an equation of six dimensions having the co-efficients of the 2d and 3d term so related that they can both be taken away at one operation. (2.) The limiting equation must always have as many possible roots as the original wanting one. Hence prove that if m consecutive terms be wanting in an equation, it cannot have more than (n-2m.) possible roots. How many possible roots can the equation a - ax2 + b = o have? (3.) Give Cardan's method for the solution of a cubic equation. Shew that it fails when all the roots are real, and succeeds when two roots are imaginary, or when all real but two equal. (4.) If several roots of an equation lie between two consecutive integers, how may Sturm's Theorem be applied to find an approximation to each? Find by this method an approximate value of a root of the equation x3-x2- 5 — v. Correct to three places of decimals. (5.) Explain Newton's method of approximating to the roots of an equation, and shew when it may safely be applied. = Obtain an approximate value of a root of x3 + 4 x2-1 o. Correct the two places of decimals. (6.) Define the asymptotes of an hyperbola. If any straight line Qq perpendicular to either axis of an hyperbola meet the asymptotos in Q and y and the curve in P, the rectangle QP. Pq is invariable. (7.) In the Ellipse the sum of the squares of the conjugate diameters is constant (C P2 + C D2 = À C2 + B C.) If the normals at P and D intersect in K, shew that KC is perpendicular to P D. (8.) If any chord AP through the vertex of an hyperbola be divided in Q so that AQ: QPA C2: B C2, and QM be drawn perpendicular to the foot of the ordinate MP, shew that Q O at the right angles to Q M cuts the transverse axis in the same ratio. Appendix C. FOURTH CLASS. EUCLID AND ALGEBRA. Afternoon Paper. (1.) Upon stretching two chains, AC, BD, across a field ABCD, I find that BD and AC make equal angles with DC, and that AC makes the same angle with AD, that BD does with BC. Hence prove that AB is parallel to CD. (2.) Determine the regular polygons which by juxta-position may fill space about a point, all of them being situated in the same plane. What advantages arise from the honeycomb consisting of hexagonal cells. (3.) ABC is an equilateral triangle; E, any point in AC; in BC produced take CD CA, CF-CE; AF, DE, intersect in II. = (4.) If three clocks were regulated to go in the following manner; being set at 12 o'clock at noon on the first of January 1852; the first to keep the exact time, the second to gain a minute, and the third to lose a minute per day; what day, month and year would they meet again at the same hour. (5.) Shew how to transform a number from one scale of notation to another. Having given 16:34 in the octenary scale and 0545 in the senary, find their product in the undenary scale. Find the area of the rectangle 4 yards, 1 foot, 2 inches long, 3 yards, 2 feet, 4 inches wide. (6.) Find the sum of the series. mn + (m1) (n−1) + (m—2) (n − 2) + . . . . . . Hence find the number of balls in an incomplete rectangular pile, of 22 courses, which contain 68 balls in the length and 44 in the breadth of the bottom row. (7.) Expand at in a series ascending by powers of x. 1 + &c. to infinity is convergent, and that its limit cannot 1 (8.) An urn contains 20 balls, 4 of which are white, If a person draw 5 at a venture, find (1.) The probability of drawing only one white ball. (2.) The probability of drawing at least one white ball. (9.) If the terms of the expansion (a + b)" be multiplied respectively by the quantities m- 1 m 2 and m be a whole number, find the sum of the resulting series. r 1' (10.) Find the present value of a scholarship of Rs. 40 per month (payable monthly), the enjoyment of which is to commence 5 weeks from this date, and to continue for 12 months, at 5 per cent. simple interest. (11.) A railway train, after travelling for one hour, meets with an accident which delays it one hour, after which it proceeds at ths of its former rate, and arrives at the terminus 3 hours behind time; had the accident occurred 50 miles further on, the train would have arrived one hour and twenty minutes sooner; required the length of the line. FIRST CLASS. Morning Paper. (1.) Define a pencil of rays, converging rays, diverging rays, and the focus of a pencil of rays. If diverging or converging rays be reflected at a plane surface, the foci of the incident and reflected rays are on contrary sides of the reflector, and equally distant from it. Why does the common looking glass give more than one image at a point? (2.) Find the geometrical focus and aberration for a pencil of rays converging to a given point between the centre and principal focus of a convex mirior, and shew that, whether the rays be divergent or convergent, the aberration is towards the mirror. (3.) A small pencil of rays is incident obliquely on a concave refracting surface; find the positions of the focal lines, and show for what values of u the primary focus is further from the surface than the secondary, drawing the requisite figures. (4.) Find the deviation of a ray after two successive reflections at plane mirrors inclined to each other at a given angle, the course of the ray lying in a plane perpendicular to their line of intersection. What What must be the first angle of incidence that at a third reflection the course of the ray may be exactly reversed? = (5.) If a ray of light passes through a glass prism, shew that it is bent towards the thicker part of the prism, and that the deviation (u-1)r when the reflecting angle r, and the angle of incidence are both small. Hence deduce the position of the principal focus of a double convex lens. (6.) Find the principal focus of a refracting sphere. How may a sphere be used as a microscope? (7.) What is the dispersive power of a transparent medium, and how is it measured? What is a table of dispersive powers? Give a short account of irrationality of dispersion, and secondary and tertiary spectra. (8.) Having given two concave mirrors and two convex lenses, the focal length of the former being 4 feet and 4 inches, and of the latter 3 inches and 1 inch respectively, construct a Gregorian telescope with Huyghen's eye-piece, and find the magnifying power. (9.) Explain what is meant by a lens equivalent to a system of lenses. Two lenses whose focal lengths are 37 and 7, have a common axis, and are separated by an interval 27; if the axis of a pencil of rays crosses the axis of the lenses at a distance=1207 from the first, determine the focal length of the equivalent lens, and compare its effect with that of each of the lenses taken singly. (10.) In the simple astronomical telescope shew when the apertures of the two lenses are proportional to their focal lengths, the field of view (as seen by single pencils) is a single point. If the simple astronomical telescope be adjusted to an ordinary eye, what change must be made to suit a short-sighted person? Appendix C. SECOND CLASS. HYDROSTATICS AND SPHERICAL TRIGONOMETRY. Morning Paper. (1.) What is the principle of the transmission of fluid pressure? How far is it necessary to prove it by experiment? When a body is immersed in a fluid, prove that the pressure of the surrounding fluid acts every where in a normal to the surface. (2.) Explain the phenomena of reciprocating springs, and show that they will not reciprocate in very wet or very dry weather. (3.) The surface of a fluid at rest is a horizontal plane. If a vessel be filled with oil and water, explain why they will not mix, and show that their common surfaces will be horizontal. (4.) Find the pressure of a fluid upon any plane surface immersed in it, and the point of application of the single resultant force. Compare the pressure on the side and on the base of a regular tetrahedron (or solid bounded by four equilateral and equal triangles) when immersed in a fluid. (5.) A body floats in water; find the condition of equilibrium. A cylinder with its axis vertical floats in two fluids of different densities; find the ratio of two parts into which the cylinder is divided by the common surface of the two fluids. (6.) Describe Nicholson's Hydrometer, and the mode in which it is used in practice. (7.) Describe the process of filling and graduating a mercurial thermometer. Are the lowest points the same under all circumstances? What point in Reaumur's and in Centigrade scale correspond to 44° Fahrenheit. (8.) The sum of the angles of a spherical triangle is greater than two right angles, and less than six. Show that the angles at the base of an isosceles triangle are equal. (9.) Express the cosine of an angle of a spherical triangle in terms of the cosines and sines of the sides. (10.) Prove Napier's rules for the solution of a right angled triangle when one of the sides is the middle part. Having given one side and an angle opposite to it, solve the triangle, and explain whether there is any ambiguity. (11.) Given the angles of a spherical triangle, shew how to find its area. TAIRD CLASS. Morning Paper. (1.) How is force estimated in Statics? A horizontal prism or cylinder will produce the same effect as if it were collected at its middle point. (2.) If several forces in the same plane tend to turn a body round a fixed point, and keep |