1. Given the three angles of a spherical triangle, to find its surface.

2. In the solution of right-angled spherical triangles by Napier's rules, what cases are ambiguous ?

3. The hypothenuse of a right-angled triangle, whether plane or spherical, being supposed invariable, to compute the corresponding variations of the two sides.

4. Prove that the sides of the polar or supplemental triangle are supplements of the angles of the given triangle.

5. In a spherical triangle, having given two angles and the included side, it is required to find the other angle.

6. Find the surface of an equilateral and equi-angular spherical polygon of n sides, and determine the value of each of the angles when the surface equals half the surface of the sphere.

7. Shew how every case of oblique spherical triangles may be solved by Napier's rules only.

8. Prove the properties of the complemental triangle; and from these properties, and the expressions for the cosine of an angle in terms of the sides, and for the cosine of a side in terms of the angles, deduce Napier's rules for the solution of right-angled spherical triangles.


9. If c be the hypothenuse of a right-angled spherical triangle,

prove that


sino = sin cos?

+ cose sin

2 10. The sum of the three angles of a spherical triangle is 1825 greater than two right angles and less than six right angles. Required proof.

11. In a right-angled spherical triangle whereof c is the 1826 hypothenuse and a and b the sides, prove that

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12. If A, B, and C be the angles, and a, b, and c the sides of a spherical triangle, and if b + c = a, prove that

sin 2B + sin 2C = 0. 13. Having given six straight lines, of which each is less than the sum of any two, determine how many tetrahedrons can be formed, of which these straight lines are the edges.

14. If A, B, and C be the angles of a spherical triangle, a, b, and c the opposite sides, and 8 the distance of a point on the surface of the sphere, equally distant from the angular points; prove that

tan? tan - 2 tan tan cos B

tan? 8

sin?B 15. Find the locus of the vertices of all right-angled spherical triangles having the same hypothenuse; and, from the equation obtained, prove that the locus is a circle when the radius of the sphere is infinite.

16. Draw through a given point in the side of a spherical 1827 triangle, an arc of a great circle, cutting off a given part of the triangle.

17. If S be half the sum of the sides of a spherical triangle,
D its area,
S S-a

S-6 S




18. By what modification may formulæ for spherical tri-
angles be adapted to plane ones?

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COS a =


19. In a spherical triangle the sines of the angles are as the sines of the opposite sides.

20. If A, B, C be the angles of any spherical triangle, and a the side opposite to A, prove that

cos A + cos B cos C

sin B sin C 21. State Napier's rules for the solution of right-angled spherical triangles, and prove the two cases in which the complement of the hypothenuse is the middle part.

22. Find an expression in terms of the sides of a spherical triangle for the arc drawn from one angle C bisecting the opposite side

C, and adapt the expression to logarithmic computation. 23. State the construction of the polar triangle, and shew that its sides and angles are respectively the supplements of the angles and sides of the original triangle.

24. Shew that in a small spherical triangle, if } of the Spherical Excess be subtracted from each of the angles, the resulting angles will be those of a plane triangle having the same sides as those of the spherical triangle.

25. Prove Napier's rules for the solution of right-angled triangles when one of the sides is the middle part; and having given one side and an angle opposite to it, solve the triangle and explain whether there is any ambiguity. 26. In a spherical triangle,

cot a sin b = cos b cos C + sin C cot A. 27. Investigate Napier's analogies : shew for what cases in the solution of spherical triangles they are applicable; shew also how these cases may be solved by the aid of Napier's rules alone.

28. In a spherical triangle, the sides of which are small compared with the radius of the sphere, having given two sides and the included angle, find the angle between the chords of those two sides.

29. Given the two sides and the included angle of a spherical triangle, required its area ; and from the expression obtained, find the area of a plane triangle in corresponding



30. The measure of the surface of a spherical triangle is the 1831 difference between the sum of its three angles and two right angles.

31. Having given two sides and the included angle of a spherical triangle, obtain the third side in a formula convenient for logarithmic computation.

32. Having given the sides of a spherical triangle, find the sine of one of its angles.

33. Having given the hypothenuse and one angle of a right- 1832 angled spherical triangle, determine the remaining angle and sides. Is there any ambiguity in the determination of the side opposite to the given angle ? 34. In a spherical triangle

.6 A + B


С tan

cot at

6 2 2




35. Having given the three sides of a spherical triangle, find the cosine of one of its angles; and having given the three angles, find the cosine of one of the sides.

36. ABC is a spherical triangle, and CD the arc of a great circle drawn from the angle C to the point of bisection of AB;

prove that

cos . AC + cos . BC 2 cos } AB x cos . CD. Shew from this expression that if ABC be a plane triangle,

AC2 + BC2 = 2AD2 + 2CD?

37. Having given two sides and the included angle of a spherical triangle, determine its area.

38. In a spherical triangle any one side is less than the sum 1833 of the two others, and the sum of the three angles is greater than two right angles, and less than six.

39. By what alteration of the circular parts are Napier's rules applicable to the solution of quadrantal triangles? In a quadrantal triangle, having given an angle and a side opposite to it, deduce all the other parts.

4. cos


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40. If an, az, az, Qg; D,, D, represent respectively the arcs
of great circles which form the sides and diagonals of a quadri-
lateral figure described upon the surface of a sphere, prove
that the cosine of the arc joining the middle points of the
diagonals is equal to
cos a, + cos 02 + cos az + cos aq:

D. D

41. Express the sine of an angle of a spherical triangle in
terms of the sides, and adapt the result to logarithmic com-

42. Having given two sides a, b of a triangle, plane or spherical, and the included angle C, to find the variation produced in A corresponding to a given small variation of C.

43. Find the relation between two sides and two angles of a spherical triangle, one of the angles being included by the sides; and thence deduce the corresponding relation in a plane triangle.

44. Explain the construction and prove the properties of the polar triangle.

45. Enunciate Napier's rules for the solution of rightangled spherical triangles ; and prove them when the middle part is one of the sides containing the right angle.

46. If a, b, be the radii of the inscribed and circumscribed spheres of a regular tetrahedron, r, the radii of spheres to which the edges, and one face and the planes of the three others produced, are respectively tangents, prove that

Nab, r Zab. 47. In a spherical triangle, given one of the sides adjacent to the right angle, and the angle opposite to it, find by Napier's rules the other parts; and shew that there are two triangles which satisfy the conditions, the unknown quantities in them being supplementary to one another.

48. If ABC be an equilateral spherical triangle, p the pole of small circle circumscribing it, Q any other point on the surface of the sphere; prove that

cos QA + QB + QC = 3 cos pA.cos pQ.

r =

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