23. Develope ing powers of x. ex 2 1 in a series involving ascend Of what use are the coefficients of this series in expressing the law of the coefficients of the series for tan. O in terms of ? 24. Enumerate, as Newton has done, the principal proofs of the truth of the theory of universal gravitation. 1. GIVEN two sides and the included angle, find an expression for the area, (1) in a plane, and (2) in a spherical triangle. 2. A straight line cuts a parabola, whose vertex is A, in two points P and Q, and its axis in 0; ordinates PM, QN, being drawn, shew that AO is a mean proportional between AM and AN. 3. The force varies inversely as (distance). A body is projected from any point in any direction, with a velocity equal to that from infinity. Find the position of the apse, and the whole angle described. 4. On a horizontal dial the angle corresponding to a second of time at 4 o'clock, is double the angle for a second at noon. Find the latitude of the place. 5. The equation to a curve is y=x Trace it; find its maximum ordinate, and its area. 6. The earth revolving round a fixed axis, shew that a body let fall from the top of a high tower will not strike the ground exactly at the foot of the tower. Between what cardinal points of the compass will the point struck be situated with respect to the foot of the tower? 7. Express the distance of a point from the earth's center in terms of the latitude. 8. A point T moves uniformly along a straight line; another point P, with three times the velocity, always moves towards T, so as to describe the curve of pursuit. Trace the curve, and shew that the path described by T from the time when the paths are at right angles till it is overtaken by P is of their distance at that time. 9. The equation to the elliptical paraboloid being ax2+by2+ab z=abc, draw a normal to it; and determine the points where this line cuts the three co-ordinate planes. Also find the solid content of a portion contained by planes parallel to the planes of xx and y z. yz. 10. Find right-angled triangles, such that all the sides shall be rational numbers. 11. If a, b, c be the sides of a plane triangle and in a series which converges rapidly when e is nearly 13. Define the moon's variation. Give Newton's construction for it, and hence shew how it varies. Monday Afternoon. Mr. WHEWell. FIFTH AND SIXTH CLASSES. 1. Find the value of .151636363, &c. of £1. 2. Find in what time at compound interest, at 5 per cent. a sum will become 10 times its original value. (N. B. the log. of 105 is 2.0211893.) 3. Solve the equations x + √ {x2+ √(x2+96)} = 11 x (y+z)=a) y(x+x)=br x(x+y)=c) x3-6x-40=0, by Cardan's method. 4. A beam rests with one end on a horizontal plane, and the other against a vertical wall; find the horizontal force necessary to prevent its lower end from sliding outwards. 5. A projectile is to be thrown across a plain 120 feet wide, to strike a mark 30 feet high, the velocity of projection being that acquired down 80 feet; find at what angle it must be projected. 6. A piece of wood weighs 12 lbs. and when annexed to 22 lbs. of lead, and immersed in water, the whole weighs 8lbs. The specific gravity of lead being 11 times that of water, find the specific gravity of the wood. 7. A cylinder whose axis is horizontal empties itself by a hole in the lowest part; find the time. 8. A trapezium has two opposite sides equal, and the other two parallel; compare the resistance upon it, when it moves in the direction of the parallel sides, and when it moves in a direction perpendicular to them. 9. Explain why all parts of the field of view of a telescope are not equally bright; and find the proportion of the bright part to the whole in the astronomical telescope. 10. Having observed the elongation of a planet when stationary, shew how its distance from the sun may be found. |