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(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 H.
EC = AC + EC (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 + (m — 1) (n − 1) + (m — 2) (n — 2) + ........ Hence find the number of balls in an incomplete rectangular pile, of 22 courses, which contains 68 balls in the length and 44 in the breadth of the bottom row.
(7.) Expand ax in a series ascending by powers of x. Shew that
1+1+ 2+1 + 1.9.3:4 + &c. to infinity is convergent, and that its limit cannot exceed 3. (8.) An urn contains 20 balls, 4 of which are white,
If a person draw 5 at a venture, find
(2.) the probability of drawing at least one white ball. (9.) If the terms of the expansion (a + b)m be multiplied respec
mm-1m - 2 tively by the quantities z r- r-2'.
5. . and m be a whole number, find the sum of the resulting series.
(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.
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 mirror, 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 shew 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 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. Why is called the power of the 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 31 and 1, have a common axis, and are separated by an interval 21; if the axis of a pencil of rays crosses the axis of the lenses at a distance=1201 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 ?
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 shew 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 shew 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 44o 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 to find its area.
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 it in equilibrium, the sum of the moments of the forces tending to turn it in one direction is equal to the sum of the moments of those tending to turn it in the other.
How does the moment of a force measure its effect to turn it round a fixed point ?
(3.) Assuming the parallelogram of forces, determine the resultant of any number of forces in the same plane acting on a point.
At any point in the circumference of a circle two equal forces act in directions passing through two fixed points on the circumference. Shew that the resultant of these forces passes through a fixed point.
(4.) Find the ratio of the power and weight in that system of pullies where each hangs by a separate string (1) when the strings are parallel, (2) when they are inclined to the horizontal bar at angles 01, 02, 03, &c., respectively.
Suppose the number of parallel strings to be 8 and 1, 2, 3, &c., inches, their respective distances from each other, find where the weight must be attached to the cross bar in order that it may be horizontal: the weights of the pullies not being taken into consideration.
(5.) Explain the term virtual velocity; and apply it to find the condition of equilibrium on the screw. Would it be applicable if there were no friction between the outer and inner screw ?
(6.) All couples tending to turn a system in the same direction, are statically equivalent whose planes are parellel and moments equal.
How are couples estimated numerically and why ?
(7.) Find the distance of the centre of gravity of the frustum of a cone from the base; a and b being the radii of the two ends, and c the altitude of the frustum. (8.) ABGC, DEF, are two horizontal levers without weight, B and F
their fulcrums, the end D
Fof one lever rests upon the A B G
end C of the other, Hk is
"DFBC and WAB (9.) A uniform rod rests on a smooth fulcrum with one end on a rough horizontal plane, shew that the extreme position in which it will rest is given by the equation
a sin 20 sin (0 + a) = 2h sin a. 2a being the length of the rod h, the height of the fulcrum above the plane, and u = tan a.
Morning Paper. (1.) Explain the principle by which the signs of the Trigonometrical lines in the different quadrants are determined ; and from this give the proper signs to the tangent, secant, and versed sine in the third quadrant.