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7. The difference between the arithmetic and geometric means of two numbers is less than one-eighth of the squared difference of the numbers divided by the less number, but greater than one-eighth of such squared difference divided by the greater number. If x, y be any two numbers, x, y, their arithmetic and geometric means, x2, y2 the arithmetic and geometric means of ≈1, y1 and so on, find major and minor limits for the difference „ — Y2
If all the sums of two letters that can be formed with any n letters be multiplied together, then in each term of the product, the sum of any r of the indices cannot exceed the number rn - r(r+ 1).
Eliminate a from the equations
(x − a) (x − b) = (x − c) (x − d) = (x − e) (x − ƒ) ; and from the same equations, with the additional relation e = ƒ, find a quadratic equation for determining the quantity e or f. Shew also that if m', m" be the values of e or ƒ, then m" m' is a harmonic mean between a
and between cm', d-m'.
11. Two triangles stand on the same base, determine in terms of the base and of the tangents of the angles at the base the distance between the vertices of the triangles.
12. Give a construction for finding the common tangents of two circles, and shew that if through the intersection O of two of the common tangents which meet in the line joining the centres of the two circles, there' be drawn a transversal meeting the circles in A, A and B, B' respectively, then (the points denoted by B, B′ being properly chosen) OA. OB'=OA'. OB is independent of the position of the transversal.
13. Shew that a triangle made to revolve in the same
direction about its three angular points in a proper order through angles double of the angles of the triangle at the same angular points respectively will resume its original position.
14. Given a pair of conjugate diameters of a conic section, find geometrically the directions of the principal diameters, (1) in the case of the hyperbola, (2) in that of the ellipse.
15. A cone whose semivertical angle is tan-1 is
enclosed in the circumscribing spherical surface; shew that it will rest in any position.
16. A string ABCDEP is attached to the centre A of a pulley whose radius is r, it then passes over a fixed point B and under the pulley which it touches in the points C and D; it afterwards passes over a fixed point E and has a weight P attached to its extremity; BE is horizontal
and DE is vertical; shew that if the system be
in equilibrium the weight of the pulley is
and find the
17. A body of given elasticity e is projected along a horizontal plane from the middle point of one of the sides of an isosceles right-angled triangle so as after reflexion at the hypotenuse and remaining side to return to the same point; shew that the cotangents of the angles of reflexion are e + 1 and e+ 2 respectively.
18. If a heavy body be projected in a direction inclined to the horizon, shew that the time of moving between two points at the extremities of a focal chord of the parabolic path is proportional to the product of the velocities of the body at the two points.
19. If a body describe an ellipse round a centre of force in the focus, shew that the sum of the reciprocals of the squares of the velocities at the extremities of any chord passing through the other focus is constant.
20. A hollow cone floats in a fluid with its vertex upwards and axis vertical; determine the density of the air contained in the hollow cone.
21. A sphere composed of two hemispheres of different refractive powers is placed in the path of a pencil of light in such manner that the axis of the pencil is perpendicular to the plane of junction and passes through the centre; determine the geometrical focus of the refracted pencil.
22. Altitudes of the same heavenly body are observed from the deck of a ship and from the top of the mast the height of which from the deck is known; find the dip of the horizon and the true altitude.
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