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

8.

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).

9.

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'.

m', b

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.

1

15. A cone whose semivertical angle is tan-1 is

√2

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

distance AB,

5 P

and find the

2

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.

An Elementary Course of Mathematics.

Designed principally for Students of the University of Cambridge.
Third Edition. 8vo. 188.

An Appendix to the First Edition of an Elementary
Course of Mathematics.

Containing the principal Alterations and Additions introduced in the Second Edition. 8vo. 18.

Elementary Chapters in Astronomy

From the "Astronomie Physique" of Biot. 8vo. 38. 6d.

A Defence of Certain Portions of an

66

Elementary Course of Mathematics.”

In reply to the Remarks on the same, in Dr Whewell's recent Work on

66

Cambridge Education." 12mo. 6d.

Parish Sermons.

12mo. cloth, 68.

Plain Thoughts concerning the meaning of Holy Baptism.

Price 1s.

Confirmation Day.

Price 6d.

Solutions of Goodwin's Collection of
Problems and Examples.

By the Rev. W. W. HUTT, M.A., Fellow and Sadlerian Lecturer of Gonville and Caius College. 8vo. 88.

Preparing for Publication.

Elementary Mechanics,

Chiefly for the Use of Schools. By the Rev. HARVEY GOODWIN, M.A.

PUBLISHED BY

JOHN DEIGHTON, CAMBRIDGE.

Lectures on Practical Astronomy. By the Rev. J. Challis, M.A., F.R.S., F.R.A.S., Plumian Professor of Astronomy and Experimental Philosophy in the University of Cambridge.

A Collection of Problems in Algebraic Geometry of Two Dimensions. By WILLIAM WALTON, M.A., Trinity College.

Correspondence of Sir Isaac Newton and Professor Cotes, including Letters of other Eminent Men, now first Published from the Originals in Trinity College Library; together with an Appendix, containing a variety of other Unpublished Letters and Papers of Newton's. With Notes and Synoptical View of Newton's Life, by the Rev. Jos. EDLESTON, M.A., Fellow of Trinity College, Cambridge. 8vo. 10s.

Geometrical Illustrations of the Differential Calculus. M. B. PELL, B.A., Fellow of St John's College, Cambridge. 8vo. 2s. 6d.

By

A Treatise on the Dynamics of a Particle. By ARCHIBALD SANDEMAN, M.A., Fellow and Tutor of Queens' College, Cambridge. 8vo. 8s. 6d.

On the Lunar and Planetary Theories; the figure of the Earth; Precession and Nutation; the Calculus of Variations; and the Undulatory Theory of Optics. By G. B. AIRY, Astronomer Royal. Third Edition. 8vo. 15s.

The Propositions in Mechanics and Hydrostatics, which are required of Questionists not Candidates for Honors, with Illustrations and Examples, collected from various sources. By A. C. BARRETT, M.A. 8vo. 7s.

Optical Problems. By A. C. CLAPIN, Bachelor of Arts of

St John's College, Cambridge, and Bachelier-es-Lettres of the University of France. 8vo. 4s.

Solutions of the Geometrical Problems proposed at St John's College, Cambridge, from 1830 to 1846, consisting chiefly of Examples in Plane Co-ordinate Geometry. With an Appendix, containing several general Properties of Curves of the Second Order, and the determination of the Magnitude and Position of the Axes of the Conic Section, represented by the General Equation of the Second Degree. By the Rev. THOMAS GASKIN, M.A., late Fellow and Tutor of Jesus College, Cambridge. 8vo. 12s.

Solutions of the Trigonometrical Problems proposed at St John's College, Cambridge, from 1820 to 1846, By the Rev. T. GASKIN, M.A., 8vo. 9s.

Examples on the Processes of the Differential and Integral Calculus. By the Rev. D. F. GREGORY. Second Edition, edited by WILLIAM WALTON, M.A., Trinity College. 8vo. 18s.

A Treatise on the Dynamics of a Rigid Body. By the Rev. W. N. GRIFFIN, M.A., St John's College, Cambridge. 8vo. 6s. 6d.