inclined to the horizon of a given place, that the diurnal path of the shadow of the extremity of an object, erected perpendicular to the horizon, may be a parabola on the plane: the sun's declination being given, and supposed invariable during the course of one day.

1802.

Morning Problems.-Mr. Palmer.

FIRST AND SECOND CLASSES.

1. Given the three angles of a plane triangle, and the radius of its inscribed circle, to determine its sides.

2. The specific gravities of two fluids, which will not mix, are to each other as n: 1, compare the quantities which must be poured into a cylindrical tube, whose length is a (a) inches, that the pressures on the concave surfaces of the tube, which are in contact with the fluids, may be equal.

3. Determine that point in the arc of a quadrant from which two lines being drawn, one to the center and the other bisecting the radius, the included angle shall be the greatest possible.

4. Required the linear aperture of a concave spherical reflector of glass, that the brightness of the sun's image may be the same when viewed in the reflector and in a given glass lens of the same radius. 5. Determine the evolute to the logarithmic spiral.

6. Prove that the periodic times in all ellipses about the same center are equal.

7. The distance of a small rectilinear object from the eye being given, compare its apparent magnitude when viewed through a cylindrical body of water with that perceived by the naked eye.

and

8. Find the fluents of the quantities

hi y.a+y2

dr

x.a2-x2,

9. Through what space must a body fall internally, towards the center of an ellipse, to acquire the velocity in the curve?

10. Find the principal focus of a globule of water placed in air.

11. Determine, after Newton's manner, the law of the force acting perpendicular to the base, by which a body may describe a common cycloid.

12. Find the area of the curve whose equation is xy=a*.

13. What is the value of the quantity q so that force period-qx radius of circle.

14. Two places, A and B, are so situated that when the sun is in the northern tropic it rises an hour sooner at A than at B; and when the sun is in the southern tropic it rises an hour later at A than at B. Required the latitudes of the places.

15. From what point in the periphery of an ellipse may an elastic body be so projected as to return to the same point, after three successive reflections at the curve, having in its course described a parallelogram.

MISCELLANEOUS QUESTIONS.

Trisect a right angle.

Find the diameter and diagonal of a cube whose side = 7.

Similar triangles are to each other in the duplicate ratio of their homologous sides.-Prove it.

Prove the truth of the Arithmetical rules for the addition, subtraction, multiplication, and division of fractions.

Investigate the rule for the extraction of the square root.

3

Which is greatest, the ratio :√2, or 3 :3√5

Shew that the magnitude of an object to the eye varies inversely as the distance.

Find (n) arithmetic, and (p) geometric means between (a) and (b.)

Shew the method of measuring the circumference of the earth, and how to draw a meridian line.

Let a solid whose specific gravity=4, float in a fluid whose specific gravity= 10. Required the proportion between the parts immersed and extant.

Divide x, by x-a1 and by a—, to five

terms in the series.

Divide,48624097 by 179; and 14 by,7854.
Required the value of ,583 of a pound.

Quere the reason for pointing every 2d figure in extracting the square root, and every 3d in the cube

root.

Assign the physical cause of the blue appearance of the sky on a clear day, and its western redness at

sun-set.

Why are natural days unequal?

From whence does it appear that the systems of Ptolemy and Tycho are false?

Given the periodic times of the seven primary planets, to find the time between their all being in conjunction together.

Demonstrate that every angle ∞

Arc.

Radius.

Prove, that in any lever, P: W :: perpendicular from the center of motion, to the direction in which the weight acts: do. in which the power acts; and that when P and W act parallel, the lever will rest in any position, except the center of gravity be below the point of suspension.

px2

+9x—r—o of fractions.

Clear the equation 23

m

n

In how many years would (a) pounds amount to (p) pounds at (r) rate of compound interest.

Find the sum of the circulating decimal ,99999.

Given the altitude of the mercury in the barometer at the top and bottom of a mountain, to find its height.

=

his length.

Shew, that in a plane reflector, a man will see his whole length, if the reflector be Find the focus of the reflected surface.

rays of a concave

Find the sum of the series 1-+-+ &c.; and ten terms of the series 3+7+11 &c.

Shew that a ray of light issuing from a point and reflected to the eye describes the shortest path possible, when the angle of incidence the angle of reflection.

Let a body be supported upon an inclined plane, first. by a power parallel to the length, then by a power parallel to the base. Compare the pressure.