OF THE MATHEMATICAL REPOSITORY, BY THOMAS LEYBOURN, VOL. IV. London: PRINTED AND SOLD BY W. GLENDINNING, no. 25, HATTON GARDEN, 1819. I. QUESTION 331, by Mr. JOHN HYNES, Dublin. To find any number of squares whose sum and product are equal. II. QUESTION 332, by Mr. JOHN HYNES. To find two fractions such, that the sum and sum of their squares shall both be rational squares; and either of them being added to the square of the other shall make the same square. III. QUESTION 333, by JUNIUS. Required the general value of x in the equation x2-23y2—1 ? IV. QUESTION 334, by Mr. W. CALLOW. There are two such quantities that the sum of their squares exceeds their sum by a, and that the sum of their fourth powers together with their sum, exceeds twice the sum of their cubes by b It is required to find them without resolving any equation higher than a quadratic ? V. QUESTION 335, by Mr. CALLOW. ry+s=o, Having given y++qy2 + ry + s=o, and y(x + m) = 0: It is required to assign such a value to m, as will enable us to determine x and y, and thus to resolve a general biquadratic, supposing the solution of a general cubic to be known? VI. QUESTION 336, by AIEYOYO. Demonstrate the following Theorems : 1. If three circles be situated any how in a plane, and through the centres of every two a circle be described to touch the remaining one, the lines joining the centre of each of the circles with the point in which the circles passing through it meet each other intersect in the same point. 2. If there be three circles situated any how in a plane, and if through the centre of each, a circle be similarly described to touch the other two, then lines joining the centre of each of the circles with the intersection of those two which touch it, (taking always the corresponding intersections) will meet in the same point. 3. If through the centre of each of three circles situated in the same plane, tangents be drawn to the remaining two, either both to that part of the circumferences which is interior with regard to the three circles, or that which is exterior: then lines. A |