MATHEMATICAL REPOSITORY.

MATHEMATICAL QUESTIONS.

To be answered in Number XVIII. Vol

I. QUESTION 411, by THEODOSIUS.

Having given the three sides of a spherical triangle, it is required to find, in terms of these sides, the radii of the inscribed and circumscribing circles and the distance of their centres.

II. QUESTION 412, by GEOMETRICUS.

In a given polygon to inscribe another of the same number of sides, having its angles respectively equal to given angles, and either its perimeter or area a given magnitude.

III. QUESTION, 413 by ANALYTICUS.

Two vessels capable of containing a and b gallons are filled with mixtures of wine and water, in given proportions for each vessel; two equal measures, c gallons each, are filled at the same time, namely one from each vessel, and then emptied into the other. Supposing this operation to be repeated n times, what would then be the proportion of wine and water in each vessel ?

IV. QUESTION 414, by GALLICUS.

Find the position of a plane so that if a given triangle be projected orthographically upon it, the projection may be a triangle similar to a given triangle.

V. QUESTION 415, by MECHANICUS,

The chord of an arc described by the lower extremity of a simple pendulum being given in length, it is required to determine the length of the pendulum, when its vibrations through this arc are performed in the least time.

VI. QUESTION 416, by MECHANICUS.

If the point of suspension of a simple pendulum, instead of being fixed, be constrained to move along a straight horizontal line with a given uniform ve. locity, it is required to assign the nature of the curve described by its lower extremity, and determine the other circumstances of its motion.

VII. QUESTION 417, by ASTRONOMICUS.

Knowing the declination of a star, and the time of its rising or setting for any determinate place on the globe, it is required to determine the correction which must be made to give the time of rising or setting for any other place near to the former.

VIII. QUESTION 418, by EDINBURGENSIS.

An obelisk of a given height stands on an inclined plane given by position: What is the locus of that point on the plane at which a statue of a given height on the top of the obelisk subtends a given angle?

IX. QUESTION 419, by PALABA.

The distance of the centre of gravity of the surface of a certain solid from the vertex is equal to half the abscissa: determine the nature of the curve by the revolution of which round its axis the surface was generated.

X. QUESTION 420, by PROTEUS.

It is required to inscribe a triangle in a given circle, such, that the ratio of two. of its sides shall be given, and the rectangle of the segments of the other side, made by a perpendicular, from the opposite angle shall be the greatest possible.

XI. QUESTION 421, by SCANDINIANUS.

Suppose two straight lines and a point without them to be given by position; two points, given by position, may be found, such, that if any two parallels be drawn through them, to meet the lines given by position, one of the diagonals of the trapezium thus formed shall pass through the other given point.

A

XII. QUESTION 422, by LAPUTIENSIS.

It is required to draw two right lines, from the focus of a given conic section, to terminate in the curve, so that these lines shall have a given ratio to each other, and shall also include a given angle.

XIII. QUESTION 423, by C. B.. In the circle of the nth order whose equation is a in +3 = a C being taken, and the abscissa and ordinate OF and CF being drawn, if the tri, any point angle OCF is placed in the situation OEB, then the point B shall be situated in the curve, and the sum of the two arcs DB and DC shall always be constant.

XIV. QUESTION 424, by Mr. CUNLIFFE, R. M. C.

Given the segments of the base, made by a perpendicular from the vertical angle, and the rectangle of the other two sides, to construct the plane triangle by a method purely geometrical.

XV QUESTION 425, by Mr. CUNLIFFE, R. M. C.

If circles are described upon the sides of a plane triangle as diameters; and perpendiculars drawn from the angles to the sides; the segments of these perpendiculars, intercepted by each le, and the circumference of its respective circle, will be equal to each other: r quired the demonstration?

XVI. QUESTION 426, by LAPUTIENSIS.

Suppose two given weights, appended at given points, to a flexible line of a given length, considered without weight and fastened at the ends to two given points in the same horizontal line, and imagine a third given weight, appended to the line, between the other two, but at liberty to slide freely thereou by means of a ring; it is required to determine the position of the ring, and the points in the line, to which the other two weights are appended, when at rest, in the above mentioned circumstances.

XVII. QUESTION 427, by C. B.

XX. PRIZE QUESTION 430, by MECHANICUS,

If two bodies, connected by an inflexible and inextensible line, be placed on an horizontal plane, and one of them be constrained to move in a straight line given by position, with a uniform velocity: it is required to determine the circumstances of the motion of the other body, and the nature of the curve it describes.

Solutions to these questions must come to hand (post paid) by the first day of December, 1819.

MATHEMATICAL REPOSITORY.

MATHEMATICAL QUESTIONS.

To be answered in Number XIX. Vals

I. QUESTION 481, by PALABA.

A and B travelled on the same road and at the same rate from H to L. At the 50th mile stone from L, A overtook a drove of geese, which were proceeding at the rate of three miles in two hours; and two hours afterwards met a stage waggon, which was moving at the rate of nine miles in four hours. B overtook the same drove of geese at the 45th mile stone and met the same stage waggon exactly forty minutes before he came to the 31st mile stone. Where was В when A reached L?

II. QUESTION 432, by PALABA.

The curve AVR and semicircle APB have the same abscissa, the ordinate MV tan arc AP: prove that the area AMV = twice the segment AP.

P

III. QUESTION 433, by PALABA..

A parabola revolves round its axis, which is vertical, in a given time, and the angular motion will just prevent a body at any point of the curve from descending: required the parameter of the parabola?

IV. QUESTION 434, by PALABA.

To divide a given arc (A), less then a quadrant, into two such parts (p) and (9) that tanpX tan q may max.

V. QUESTION 435, by G. V.

A normal at any point of an equilateral hyperbola is equal to the distance of that point from the centre. Required the demonstration.

VI. QUESTION 436, by G. V.

C is the centre of an ellipse and F either focus, PH a tangent at P: draw the diameter PCp and pF meeting PH in H. Prove that pH the transverse dia

meter.

VII. QUESTION 437, by G. V.

P is any point in the diameter of a circle and PB a perpendicular to the diameter: draw any chord PA, and a tangent AB meeting PB in B, and draw BD and CE (C the centre) perpendicular to PA; then PE = DA. Required a demon

stration.

VIII. QUESTION 438, by PROTEUS.

Within a given triangle suppose another triangle to be inscribed, by joining the middle points of its sides; and again, within this triangle, suppose another triangle to be inscribed, by joining the middle points of its sides, and so on ad infinitum: What will be the limit of the aggregate, of the sum of the squares of the sides of all the triangles so formed?

IX. QUESTION 489, by Mr. CUNLIFFE.

Let any right line be drawn through the focus of a given conic section, ter minating in the curve; then a fourth proportional to the whole line and the two segments thereof, made by the focus, will always be of the same constant length. Required a demonstration.

X. QUESTION 440, by

A triangle being given, it is required to describe three circles, so that each circle shall touch the other two and a side of the triangle at the point of bisection.

XI. QUESTION 441, by

Required the curve that has at each point the radius of curvature a fourth proportional to the abscissa, the ordinate, and a given straight line.

XII, QUESTION 442, by PALABA.

To determine the nature of the curve such that the perpendicular from a given point upon the tangent shall be a mean proportional between a given line and the segement of the axis intercepted between the tangent and this same given point.

XIII. QUESTION 443, by PALABA.

To determine the equation to the curve whose tangent is a mean proportional between the segment of the axis intercepted between it and a given point, and that same segment augmented by a given line.

XIV. QUESTION 444, by PALABA.

A body, urged by a force perpendicular to the horizon, describes the quadrant of a circle. Required the law of force which will make it recede uniformly from the horizontal radius, and the time elapsed and the velocity acquired at any point of the descent.

XV. QUESTION 445, by

The characteristic property of the circle is, that all the chords which pass through a certain determinate point in its plane are equal: but there exists an indefinite number of curves which possess this property. It is required to find the most general equation to curves of this nature.

XVI. QUESTION 446, by Mr. CUNLIFFE.

A given rod or beam, has one end suspended by a cord of a given length, fixed at a given point above an inclined plane of a given inclination; it is required to determine the position of the beam, weight sustained by the chord, and pressure against the inclined plane, when the beam is in equilibrio.

XVII. QUESTION 447, by G. V.

Find the rectification and quadrature of the magnetic curve, which is such, that if lines be drawn from the poles to any point in the curve and and be the angles they make with the line joining them, cos + cos = c, a constant quantity.

XVIII. QUESTION 448, by ERATOSTHENES.

Required a number consisting of six digits abcdef such that, its multiples by 1, 2, 3, 4, 5, 6 may contain amongst them the following arrangements:

XIX. QUESTION 449, by ERATOSTHENES.

It is well-known that the formula x2 + x + 41 contains a great number of primes: prove that when a is of either of the two following forms it cannot be a prime, x = 53 a + 8 and a 97 a—

- 8.

XX. PRIZE QUESTION 450, by MECHANICUS.

If the beam in question 401 instead of resting upon the curve, be at liberty to slide freely over the prop and descend by the force of gravity, it is required to determine the motion of its centre of gravity.

Solutions to these questions must come to hand (post paid) by the first day n June, 1820.

MATHEMATICAL REPOSITORY,

VOL. IV. PART I.

MATHEMATICAL QUESTIONS.

ARTICLE I.

Solutions to Questions proposed in Number XII.

I. QUESTION 331, by Mr. JOHN HYNES, Dublin. To find any number of squares whose sum and product are equal.

SOLUTION, by Mr. CUNLIFFE, R. M. College.

In order to make the solution quite easy, I shall begin with finding two squares whose sum and product shall be equal to each other; and afterwards proceed gradually to finding three, four, and five squares, whose sum and continual product shall be equal to each other, till the way of extending the solution to any number of squares, is sufficiently clear and evident.

1. To find two square numbers whose sum and product shall be equal to each other.

Let x and y denote the two squares; then, by the question,