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(B.) Solve the fluxional equation sijo– yžy – = 0.
10. If A + B be less than a semicircle,
A + B
:: rad. : sin. 2
2 11. If P be put for the semi-perimeter of a spherical triangle, the sides of which are denoted by a, b, c, and the opposite angles by A, B, C,
sin. P sin. (P-a) cos. } A =
sin. b sin. c 12. The upper extremity of an inclined plane being given, to determine its position, so that the time shall be a minimum, in which a body falls down it, and afterwards moves to a given point in the horizontal plane, with that part of its acquired velocity, which is not destroyed by its impact on the horizontal plane.
13. A given sphere, and its circumscribing cylinder, of the same uniform density, being supposed to revolve round their axes, with equal angular velocities, to compare their momenta.
14. A hollow sphere is to be formed of a substance, the specific gravity of which is greater than that of air, in the ratio of n to 1, and is afterwards to be filled with gas, the specific gravity of which is less than that of air, in the ratio of 1 to m; the thickness of the shell being given, to find its diameter so that it may float in the air.
15. To describe the construction, and determine the magnifying power of a Compound Microscope.
16. To describe the construction of an Achromatic Lens, and explain the reasons of that construction.
17. To determine the Sun's parallax, from observations made on the transit of Venus.
18. The times of a star's transit over the meridian, and over two vertical circles at given distances from the meridian, having been observed, to compute the latitude of the place of observation, in terms of the azimuths, and hour-angles thus given.
19. To determine under what circumstances of the velocity of projection, a body, projected from a given point, in a given direction, and acted upon by a force inversely proportional to the mth power of the distance from the centre, will come to the centre, or to an apse.
20. If a body, acted upon by the constant force of gravity, fall down the concave side of a circular arch, the tangent of which, where the body begins to fall, is perpendicular to the horizon, to find the point where the pressure on the curve shall be equal to an nth part of the weight of the body.
1. Sold one-half of a bale of goods for £50. and by so doing lost five per cent. At what price must the other half be sold, so that five per cent. shall be gained on the whole ?
X% 2. Solve the equation
+ 20% = 423.
2 3 3. Similar triangles are in the duplicate ratio of their homologous sides.
4. Let a, b, y, be the respective angles, which a line makes with three rectangular co-ordinates; shew that these angles are so connected that
cos.' a + cos.' B + cos.'y = l. 5. In a plane triangle, whose angles are A, B, C, and the opposite sides a, b, c, the angle A may be expressed by the series
mo m. sin. C+
sin. 2C +
sin. 3C, &c. 2
3 m being the value of the fraction o
7 6. Find an expression for the excess of the sum of the three angles of a spherical triangle above two right angles.
7. In the revolution of a body round the focus of an ellipse of small excentricity, show that the supposition of the angular velocity round the other focus being uniform is not strictly accurate.
8. Find the centre of gravity of the surface generated by a common semicycloid revolving round the tangent at the vertex as an axis.
1 9. Find the sum of series
to a terms;
1 and of the series
&c. ad infinitum. 1.3 3.5 5.7
i 10. Find the fluent of
And solve the equa
JA + Bar - C. tion
xi + yj – yi + xy = 0. 11. AP is a portion of the common parabola, PT a tangent at the point P, PB the normal, and to a perpendicular to the axis
from the intersection of the tangent. Show that if BP be produced to meet TO in O, BO is equal to the radius of curvature at the point P. 12. In throwing cross and pile, it is known that the piece has a
1 tendency one way, estimated at 20. Required the probability of throwing cross twice together.
13. Find the angle between the true and apparent path of a Star affected by refraction during the time of an observation.
14. Find the position, in which a given isosceles prism will float in a fluid, the vertex being immersed.
15. The oscillations of a pendulum in a medium when the resistance varies as the velocity are isochronous. Compare the time of an oscillation with the time of an oscillation of the same pendulum in vacuo. 16. Construct the curve whose equation if y*
96 aʼy? + 100 aʼx? x+ - 0. Show whether there be any points of inflexion,
or any asymptotes, and find the greatest values of x and y. 17. Trisect a circular arc, and construct the equation either by the conchoid of Nicomedes, or by means of the hyperbola.
18. Show that the sine of incidence is to the sine of refraction in a given ratio, both in Huygens' hypothesis of Undulations, and in Newton's of Material Particles attracted by the medium through which the rays of light pass.
Investigate the velocities of the ray before and after refrac
tion on both hypotheses. 19. Explain the method of Exhaustions, by which the ancients were enabled to measure the periphery of the circle.
20. Find the horary motion of the Moon's nodes in a circular orbit, (Newton, B. III. Prop. 30.)
1. A GROCER had 150lbs. of tea, of which he sold 50lbs at 9s. per pound, and found that he was then gaining only 71 per cent. But he wished to gain 10 per cent. on the whole. At what rate must the remaining 100lbs, be sold that he may attain his wishes ?
2. Solve the following equations:
x + y +
= 140 x2
X = 3
y Ny = 4 (3.) **
· 5x3 + 7x2 5x 6 = 0. (4.) x = a (by approximation). 3. In what time will a sum of money placed at compound interest double itself at 4 per cent. ? 4. Required the value of the infinite series 718 + 189
V18 + 189
718 + &c. 5. Required the value of y in the following equations, when sel. (1.) y=(1 - x), tan. ho (* = semi-circumference of a
2 circle, rad. = 1).
2 (2.) y=
1-3+log.x 6. Transform the equation 2 5x + 8x - 5= 0 into one whose roots are the squares of the differences of every pair of roots : and show the mode of determining, from the transformed equation, the impossible roots in the original equation.
7. From a bag containing four white and eight black balls, three persons (A, B, and C.) take each a ball in turn, viz. A first, then B, Then C, and so on in succession, until the person, who first drawsa white ball, wins. What are their respective chances ?
8. In a spherical triangle, the two sides and the angle between them being given, find the base.
9. The vertical angle of an isosceles spherical triangle is always greater than the angle included between the chords of the equal sides. 10. Sum the following series : (1.) 2 + 6 + 12 + 20 to n terms. 1 1 1
&c. ad infin. 1.2 4.5 7.8 10.11 (3.) Cos. A + cos 2A + cos. 3A + &c. to n terms. 11. Sum the series, sin. o x sin. 6 + sin, 20 % sin, 20 &c. to s
terms: and show that when n is infinite, the sum = 0 whatever be the ratio of p to 0, except that of equality.
12. A body attracted towards a centre by a force varying inversely as the square of the distance from the centre, meets at a given point of its rectilinear descent with a plane inclined at an angle of 45°. Required the time from the beginning of motion to its reaching the centre.
13. A cylindrical wheel, whose weight is P, unwinds itself from a string passing round its circumference, what weight (W) attached to the extremity of the string will be kept at rest on a plane of given elevation as P descends vertically?
14. A sphere and its circumscribing cylinder revolve round their common axis. Required the ratio of the momenta generated in a given time.
15. An homogeneous circular wheel vibrates edgeways being suspended from a point in the circumference. Required its centre of oscillation.
16. Find the centre of gravity of the area included by the arc of a cycloid, by a tangent at the vertex, and by two rectangular ordinates equi-distant from the vertex.
17. In the common parabola the radius of curvature is equal to the cube of the normal divided by the square of the semi-parameter.
ar - x 18. Trace the curve whose equation is y=-
✓ 2ax - x2 find the angles at which it cuts the line of the abscissæ.
19. O is the centre of the circular arc AB, OBT is the secant. The exterior part BT is continually bisected in P. Required the area traced out by OP.
20. Two balls (A and B) are previous to motion at a given distance from each other in the same vertical line : from what height above the horizontal plane must A be let fall—so that B, which is perfectly elastic, may after reflection meet A at a given distance above the plane?
21. Two balls lying on an horizontal plane are connected by a string of unlimited length which passes through a ring in the plane. One of the balls is projected in a given direction with a given velocity and draws the other towards the ring. Required the curve which the projected ball describes. 22. Find the following fluents :
a sz.cos. z . cos. nz, sz.cos.'
N1 construct the fluent
when a is less than 1;
xy — yi = i V x2 - y".