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1. Define the 1st, 2d, 3d, &c., differentials of a function according to Newton's method of limits, and also according to Lagrange; shew that both methods coincide, and exemplify in the case of sin. x. 2. Prove the rules for differentiating,
(1.) where u and z are both functions of x.
(2.) tan. x.
(4.) log. x. and differentiate.
(1.) (a + bx*)";
v1 + q*+ V1 - 2* (2.)
NI + – N1 xa
(3.) log: )
✓(a+b)/ (1+cos. x) 3. Find the area (1.) of the cycloid, (2.) of the logarithmic curve, (3.) Shew that the surface and solidity of a sphere, are each two thirds of the surface and solidity of the circumscribing cylinder.
4. Deduce Maclaurin's theorem from Taylor's, and apply the former to find the tangent in terms of its arc.
5. Transform the series for log. (1 + x) into one which con
when x = 0,
verges for all values of x; and apply it to compute the modulus of Briggs's system.
6. In a series A+Bx+Cx? + D.x3...... where the ratio which any one coefficient bears to that immediately preceding it, is always finite, such a value may be found for x as will render
any one term greater than the sum of all that follow. Required a proof, and de termine whether this be possible in the series,
1 + x x+38 22 +48 x + .....
1+1.2.x+1.2.3 x +220.127.116.11.2..... 7. Shew by the Differential Calculus, or from considerations purely algebraical, that if a : x ::y:b; x+y will be a minimum when x = y.
8. Required all the values of x which will render y a maximum or minimum in the following equations, and determine those values which give maximum results, those which give minimum, and those which give neither.
(1.) y = 2 5x+ + 5.43 + 1
4ao xy + 24 = 0. 9. Prove that under a given spherical surface, a hemisphere contains the greatest segment of a sphere.
10. Required the least paraboloid that can be circumscribed about a given sphere. 11. Find the value of:
- a + (x – a)
(x2 – a?)?
2 12. Find the equations to the tangent, and normal of a curve at a given point, and apply them to the parabola.
13. A curve is convex or concave towards the axis of the abscissæ, according as the ordinate and its second differential coefficient have the same or different signs. Required a proof.
14. Trace the curve whose equation is x'yé-a'x + aʼx = 0; find its point of contrary flexure, and the angle at which it cuts the axis.
15. Draw an asymptote to the reciprocal spiral.
16. Investigate the general expression for the radius of curvature in curves referred to an axis, and apply it to the catenary whose
x + a + 2ax + x) equation is y := a. log. (
17. Find the differential of the surface generated by the revolution of a curve round its axis. 18. Find the integrals of xadx Ar’dx – B.xd.x
(2ax + x) dix (4.)
x/ (1 – ze)" VOL. II.
x+a+i (248 + 2) )
19. Find the content of the solid generated by the revolution of a cycloid round the tangent at its vertex.
20. Prove that the solid generated by the revolution of any regular polygon inscribed in a cirele round its diameter, is equal to one-third of the surface generated, multiplied into the perpendicular drawn from the centre on one of its sides.
21. Find the area of the curve whose equation is y=px, between the values of x = a, and x = b; and investigate the value of the result in the case of the common hyperbola. 22. Shew that all the curves included under the equation y=pr',
1 are rectifiable when
is a whole number.
2 (n-1) 23. Find the length of an arc of the spiral of Archimedes contained between two polar distances p and q; and supposing the radius of the circle to be a, prove it equal to an arc of a parabola contained between two ordinates p and q, the latus rectum being a.
24. Find the convex surface and solid content of the portion of a right cylinder, which is cut off by a plane passing through the centre of the base, and inclined to it at an angle of 450.
1. If two forces each (=A) act at an angle (mo) they will 1s"
2 2. A given force is to be resolved into different pairs of forces, whose sum shall bear to it the ratio of 5:3. Determine the minor axis of the limiting ellipse.
3. When a piece of timber of unequal dimensions is pat on a fulcrum 13 feet from the smaller end, it is in equilibrio; but when put on a fulcrum only 12 feet from the smaller end, it requires a weight of 210 lbs. on that end to support it. Find the weight of the timber?
4. The arms of a false balance are as m : n, and a body weighs (a lbs.) at one end, and (blbs.) at the other; find,
(1.) The true weight of the body.
(2.) The excess of the sum of the two apparent weights above
twice the true weight. (3.) The distance of the point of suspension from the middle
of the beam. 5. Suppose a billiard-table to be an irregular hexagon, and (A) and (B) two balls upon it given in position. Determine against what point of any one side the ball (A) must impinge, so that after rebounding from it, and from every other side in succession, it may hit the other ball B.
6. In the time in which a heavy body falls down a well, and its sound on the bottom returns to an ear at the top, a pendulum 61 inches long vibrates eight times :- What is the depth of the well?
7. Find the lines of swiftest and slowest descent from one given circle to another.
8. Determine in what locus a person must be placed, to throw a perfectly elastic ball against a given point in a given vertical plane, so that the ball may each time return to his hand.
9. In the collision of imperfectly-elastic bodies, find the ratio of the relative velocity before impact to the relative velocity after impact; and show that the sum of the products of each body into the square of its velocity before impact, is greater than the sum of the products of each body into the square of its velocity after.
10. What must be the form of a triangle whose centre of gravity is the centre of the circumscribing circle?
11. A and B are hung over a fixed pulley :-Required the ratio of H : L of an inclined plane on which a body shall descend as many feet in a given time as the heavier (A) of the two bodies.
12. A weight (W) is drawn along the horizontal plane (CB) by another weight (P) acting over a pulley fixed at the perpendicular height (BA) above the horizon :-With what force will (W) be accelerated ? And what will be its velocity acquired at (B)?
13. In the steelyard, if the weight increase in arithmetic progression, the divisions of the scale will be at equal intervals; and if each of these intervals be equal to the shorter arm, the moveable weight will be equal to the difference of the arithmetical progressions.
14. A body being let fall from the top of a tower, was observed to fall half way in the last second :—What was the tower's height?
15. (AB) and (AC) are two inclined planes of a common height (AD); the length of the plane AB=a:- To find what must be the length of the other plane AC, so that a given weight (P) on the plane AB may draw another given weight (W) up the plane AC in the least time possible, (P) and (W) being connected by a string over a pulley at (A).
16. A body is projected at an angle of 45° with a velocity acquired by a heavy body falling down the axis of a cycloid : Required the ratio of the time of fight to the time of oscillation in the cycloid ?
17. There is a mountain on the earth's surface of such a height: that a clock (which keeps true time at the bottom) when carried to the top loses two minutes a-day :—What is the altitude of the mountain, supposing the earth's radius to be 6982000 yards?
18. If the axis of a parabola be perpendicular to the horizon, and chords be drawn from the vertex to any point in the curve, compare the times of descent down them by the force of gravity.
19. What is the least velocity with wbich a body must be projected from the top of an inclined plane, so as just to reach the extremity of the plane ?
20. Suppose a body to be projected downwards from a given point (A) with a given velocity (a); and after (n) seconds are elapsed, another body is projected upwards from a given point (B) with a given velocity () Where will they meet?
21. What number of pulleys (in a system where each pulley hangs by a separate string, and all the strings are parallel) must be applied to a weight (=96 lbs.) so that P (=11b.) may sustain it on an inclined plane whose height is one-third of its length, P being supposed to act parallel to the length ?
22. Define the centres of gravity, gyration, oscillation, and percussion; and find with what part of a cylindrical stick 50 inches long must (B), whose arm is twenty inches long, strike (6), so as to give the greatest possible blow.
23. If a chord of a given length be fastened to two hooks, A and B, not in a horizontal line, and a weight (W) slide freely along the chord, find where (W) will rest; and compare the pressure on either hook with the weight (W).
24. A globe weighing 200 lbs. is supported between two inclined planes whose inclinations are 60° and 45° respectively :-What is the weight supported on each plane ?
25. Investigate a general Theorem for determining the ratio between (P) acting at any angle on the back of a scalene wedge, and the sum of the resistances (supposed to be wholly effective) acting at different angles on the sides ; and apply it to an isosceles wedge, when (P) acts perpendicularly, and the equal resistances act parallel to the back.