and we shall find, as above, that the whole difficulty consistsin integrating the double of the second part from p = 2 top=∞, Let p q2; we have then

If we now make m*— 12, and q* + m2 = q2y, which gives 29=√(y+2m) + √(y —

formed expression

dy

2m), we shall have for the trans

idy

√ (y + 2 m ) √ ( ) — 2m2 = 7) + √ √(y—2m) √ (y2 — 2m2 —7)

√(y2 —

the two parts of which ought to be integrated from y 2√3. Let 2√3n, y + 2m = x2,

or y=2m2 + 7 to y =∞.

the first part becomes

In like manner, if we make b-4+3, the integral of

b=

1+√3

the second part will be √3 = 1) F1b; Fib; and we may de

4

sign the second modulus by b, because, we have b2+c2 1, and so they are the complements of each other. We have therefore,

But it happens in regard to these functions, that we have exactly F1b = 3F1c, the value of [] is therefore reducible to

this []=✓ 3 − 1) F1c, and as c is very small, this function

1

may be easily valued. Hence it appears that the case of n = 12 is also resolved completely by functions of the first kind. The end of Le Gendre's paper on Elliptic Trancendentals.

ARTICLE II.

To the Editor of the Repository.

SIR,

As in one of your earlier numbers you inserted the following

x&c. I hope you will have no objection to insert the following geometrical approximation. It is given by Euler, Novi Comment. Petropol. Vol. III.-q is taken a quadrant for facility of application in practice.

I am, &c.

R. I. DISHNEAGH.

Let AB (fig. 10, pl. A'.) represent a quadrantal arc: rad. = 1, Bisect it in c, and let oc be the secant and . to secant

Bisect Ac in d and draw odd to meet CD

OD: OC:: sec COD: 1

to OC,..

bisecting ad in e and making a similar construction.

x sec 1.

4

bisecting ae in f, aƒ in g, &c. and making the same construc

tion in each case.

End of the third Part of the third Volume.

Printed by W. Glendinning, 25, Hatton Garden, London.

As there are many readers of the Mathematical Repository who would, no doubt, be glad to see the problems proposed annually at the University of Cambridge, the Editor has obtained permission from the Moderators to insert in his work those for the year 1811. And if in time coming he shall meet with the same indulgence, he will continue to insert those of future years.

It is, however, to be observed, that they are not proposed here for the purpose of being resolved.

The problems that have been proposed for the last ten years have been lately collected into a volume and published under the title of Cambridge Problems, by the Publisher of the "University Calendar.'

THE SENATE-HOUSE PROBLEMS,

Given to the Candidates for Honours during the Examination for the Degree of B. A. in January, 1811,

BY THE TWO MODERATORS,

MONDAY, JAN. 14, 1811.

MORNING PROBLEMS.-Mr. TURTON.

1. The interior angles of a rectilinear figure are in arithmetic progression; the least angle is 120°, and the common differ ence 5°. Required the number of sides.

2. Given the radii of two spheres, and the line joining their centres; Find, in that line, the position of an eye, to which the apparent surfaces will together be the greatest possible.

3. The weight of a globe in air = W, and in water — w; Find its diameter and specific gravity, having given the specific gravity of water (S) and of air (s).

4. Having given the latitude of the place, the day and hour, also the latitude and longitude of a star; Find its altitude and azimuth, the point where its vertical circle cuts the ecliptic, and the angle which they make.

5. Find the ratio of the velocity at the extremity of the latus rectum of an ellipse (the force being in the focus) to the velocity

VOL, 111.

in a circle whose radius is the distance of the nearer apside from the focus; and shew that, as the excentricity is increased, this ratio approaches to a ratio of equality.

6. Shew that the spaces described by a body, impelled from rest by a finite variable force, are, 'ipso motûs initio," in the duplicate ratio of the times.

7. If, to the radius unity, A = the sum of the tangents of any number of arcs; B = the sum of the products of every two of them: C the sum of the products of every three; and so on: Shew that the tangent of the sum of those arcs will be

9. Find the value of a. a+r. a +2r, &c. continued to any number of factors.

10. Find the nature and length of the caustic, when the reflecting curve is a circular arc, and the focus of incident rays is in the circumference of the circle.

11. At a given place, at a given hour, and on a given day, required the point of the compass on which a rainbow would

appear.

12. Given the latitude of the place, and the day of the year; Find the hour at which two stars, whose right ascensions and declinations are known, will be on the same azimuth.

12. Given the perihelion distance of a comet describing a parabola, and the radius of the earth's orbit, here supposed to be circular; Compare the time of the comet's moving through go degrees of true anomaly with the length of the solar year.

14. Define the center of spontaneous rotation of a system; explain the principle on which that center may be found; and shew that if the system revolve round an axis, passing through that center perpendicular to the plane of revolution, the former point of impact will become the center of percussion.

MONDAY AFTERNOON.

Fifth and Sixth Classes.-Mr. TURTON:

1. The interest of 251. for 3 years, at simple interest, was found to be 37. 18s. 9d, Required the rate per cent. per annum.

2. If the first of six magnitudes be to the second as the third to the fourth, and the fifth to the second as the sixth to the fourth; Prove that the first and fifth together will be to the second, as the third and sixth together to the fourth.

dz
of
Na+ bz' Va+ bz2,

xx

and of

Vax-x2

5. Find an expression for the sum of n terms of the

+ &c. that may be applied according as n is an

5 15 45

even or an odd number.

6. Shew, that if any momenta be communicated to the parts of a system, its center of gravity will move in the same manner that a body, equal to the sum of the bodies in the system, would move, were it placed in that center, and the same momenta, in the same directions, communicated to it.

7. Compare the time of oscillation in a given cycloid, with the time of falling down a vertical line equal to the whole length of the cycloid.

8. Required the equation of which the roots are +√2, 3, 4. 9. If a body fall through a finite altitude AS, the force varying inversely as the square of the distance, and on AS, a semicircle ADS be described; Prove that the area described by the indefinite radius SD is equal to the area uniformly described, in the same time, in a circle whose radius is the half of SA.

10. Given the latitude of the place, and the sun's declination; Find the length of the day.

11. Compare the time of descent through any space AS, the force at S varying inversely as the square of the distance, with the periodic time in a circle whose radius is SA.

12. Explain by what means the accelerating forces of bodies. are compared; also, by what means their moving forces; and shew that the accelerating force varies as the moving force directly, and the quantity of matter moved inversely.

13. Prove, that if the object placed before a spherical reflector be a straight line, the image is a conic section.

14. Two weights, of which one (P) is known, are connected by a string passing over a fixed pulley; P, in descending from rest through the space s, acquires the velocity a. Find the other weight.

15. Find the variation of the force by which a body describes a parabola, round a center of force in the focus.

16. Find the actual periodic time in a given ellipse, described