twice the sum of the products of these sides taken two together and multiplied by the cosine of the angle between them.

5. Find the angle at which a side of a pyramid is inclined to the base, the sides being equilateral triangles, and the base a square.

6. A vessel observed another ao from the North, sailing in a direction parallel to its own. In p hours its bearing was Bo, and in q hours afterwards

from the North. To what point of the compass were the vessels sailing?

7. Three circles, whose diameters are √3−1, √3 +1, and 3-3 respectively, touch each other in the points A, B, C. Find the area of the triangle ABC.

8. Two regular polygons of the same number of sides being described, the one within, and the other without, the same circle; what will be the number of sides when the space included between the two polygonal boundaries is of the interior polygon?

9. Coasting along shore, observed two head-lands: the first bore N.N.W., and the second N.E. by E.; then steering 12 miles in the direction E.N.E., the first bore N.W., and the second N.E. Shew how the bearing and distance of the two head-lands from each other may be found.

10. Compare the areas of decagons inscribed in, and circumscribed about, a circle.

11. A semicircle is divided into three arcs A, B, C, whose cosecants are in harmonical progression, shew that

12. A regular polygon is described in a circle, and the tangent of half the acute angle which a side subtends at the circumference = t; shew that a side of the figure the diameter of the circle :: 2t: 1 + t2.

13. Required the perpendicular from the vertex upon the base of a triangular pyramid, all the sides of which are equilateral triangles of given area.

14. A circle is inscribed in an equilateral triangle, an equilateral triangle in the circle, a circle in the latter triangle, and so on ad infinitum; if r, r1, r2, T3...be the radii of the circles, prove that

15. If a, b, c, d be the four sides of a quadrilateral figure inscribed in a circle, and 28 = a + b + c + d, and A be the angle contained by a and d, then

(S − b) (S – c),

and the area = √(S − a) (S − b) (S – c) (S′ – d).

16. The sides of a plane triangle are as 3, 5, 6; compare the radii of the inscribed and circumscribed circles.

17. If from any point O within a triangle, three straight lines be drawn from the angles A, B, C, meeting the opposite sides in a, b, c, then will

18.

In any right-angled plane triangle, twice the side of the inscribed square is an harmonical mean between the sides containing the right angle.

19. Prove that the area of a regular polygon inscribed in a circle is a geometrical mean between the areas of an inscribed and of a circumscribed polygon of half the number of sides; and that the area of a regular polygon circumscribed about a circle, is an harmonical mean between the areas of an inscribed one of the same number of sides, and of a circumscribed one of half that number.

20. If the side of a pentagon inscribed in a circle be 1, the radius is

21.

√5+√√/5

10

Given the radius of the circumscribed circle, and the three angles of a triangle; find expressions for the three sides.

22. If R, r be the radii of the circumscribed and inscribed circles of a regular polygon of n sides, and R', r' the corresponding radii for a regular polygon of 2n sides, and of the same perimeter as the former, then

23. An indefinite area can be divided into no other regular figures than triangles, squares, and hexagons.

24. If A, B, C be the angular points of a triangle, a, b, c points in the sides respectively opposite to them, prove that the lines Aa, Bb, Ce will intersect in a point, if

25. If R, r be the radii of circles circumscribed about and inscribed in the same plane triangle, prove that the distance between the centres of the circles R-2 Rr.

=

28. If a quadrilateral is capable of having a circle inscribed in it, the sums of the opposite sides are equal to one another; and if, besides, it is capable of having one circumscribed about it, its area equals the square root of the continued product of the sides.

cot ẞ - cot = cot (a + 0) + cot (a – ß).

31.

A lamp on the top of a pole 32 feet high is just seen by a man six feet in height, at a distance of 10 miles; find the earth's radius.

32. A ship, the height to the summit of the top mast of which from the water is 90 feet, is sailing directly towards an observer at the rate of 10 miles an hour; from the time of its first appearance in the offing till its arrival at the station of the observer is 1 hour 12 minutes : find approximately the earth's radius.

STATIC S.

COMPOSITION AND RESOLUTION OF FORCES.

1. THREE forces acting in the same plane keep a point at rest; the angles between the directions of the forces are 135o, 120o, and 105°; compare their magnitudes.

2. Two forces sustain each other by means of a string passing over a tack; prove that either force: pressure on tack ::: cosine of half the angle between the directions of the forces.

3. A body is suspended from a given point in a horizontal plane, by a string of known length, which is thrust out of its vertical position by a rod (without weight) acting from a given point in the plane against the body; find the tension of the string.

4. If be the angular distance of a body from the lowest point of a circular arc in a vertical plane, the force of gravity in the direction of the arc that in the direction

5. AB is a given horizontal line, BC a rod without weight moving freely in a vertical plane about B. A weight is suspended by a string fixed at A and passing over the end C of the rod. Find the position of equi

librium.

6. A string PAQ is knotted to a fixed point A, and drawn in different directions by the forces P and Q, in such

a manner that the pressure on A =

PAQ.

P+Q
2

; find the angle