37

E.—la

10. Find relations between the trigonometrical ratios of A and those of 180° —A, 180°+A, and 90°+A. Trace the changes in tan 6 as 0 increases from — w to ir, and draw graphs of y = sin x, y = cos x, and y = tan x, between the limits —ir and -ir of x. 11. In any triangle prove— (i.) tan A-f tan B + tan C = tan A tanß tanC ... sin A sin B sin C v ll -) ~ - ~~r -~r ... A-B α-b C (ill.) tan-2-=— jCotg 12. Show how to solve a triangle, given two sides and the included angle. If a = 410-03 chains, b = 48718 chains, G = 68° 33' 58", find the remaining parts. Log 7-715 = -88734 log 4-100 = -61278 log 5-093 = -70697 log 8-972 = -95289 log 4-101 = -61289 log 5-094 = -70706 log 8-973 = -95294 Ltan 55° 43'= 10-16639 Ltan 7° 11' = 9-10049 Ltan 55° 44'= 10-16666 Ltan 7° 12'= 910150 Lsin6B°33'= 9-96883 L sin 48° 31' = 9-87457 Lsin6B°34'= 9-96888 L sin 48° 32'= 9-87468

No 38. — Elementary Mechanics. — For Civil Service Junior. Time allowed: Three hours. [All answers should be illustrated, where possible, with diagrams.'] 1. Define mass, acceleration, pound, poundal, and momentum. Write down the main formulae that connect the acceleration with other magnitudes. What is the weight in poundals of a mass of 401b, at a place where the body would fall through only 12 ft. in the first second ? 2. Explain fully the principle on which the recoil of a gun depends. A quarter-ton shot is discharged from an eighty-one-ton gun with a velocity of 1,750 ft. a second : what will be the velocity of recoil of the gun ? How would it affect your answer if the mass of the powder, which also is a quarter of a ton, were taken into account ? 3. Prove (or describe an experiment arranged to verify) the parallelogram of forces. Show how you would find by construction the angle between two forces, each of 12 units, which have a resultant equal to 6 units. 4. State the conditions necessary for the equilibrium of a body free to move in one plane, and acted on only by a system of parallel forces. To what do these conditions reduce when one .point in the body is fixed ? Three parallel forces of 2, 6, 8 units are in equilibrium : find how they are disposed, and illustrate by a figure. 5. What is the C.G. of a body ? How would you verify by experiment that the C.G. of a cube is equidistant from the faces ? If a cube be placed on a plane with an edge about which it turns horizontal, at what inclination of the plane will the cube just topple over, the cube being supposed not to slide ? 6. Describe the requisites of a good balance. Supposing that a balance is accurate except for an inequality in the arms, and that a body that appears to weigh 49 pounds when placed in the one scale-pan appears to weigh 51 when placed in the other, find the true weight of the body to three places of decimals. 7. How is the pressure of a fluid measured when variable ? Find the pressure at a depth of 600 yards below the surface of the sea in pounds to the square inch, taking a cubic foot of sea-water to weigh 1,025 oz. 8. Distinguish between density and specific gravity. Describe two different methods for finding the specific gravity of a liquid. How would you find that of sand ? 9. Prove that when a body floats in equilibrium the weight of the liquid displaced is equal to that of the body. What determines the particular position in which the body will float? A body floats with one-ninth of its volume above the surface of pure water : if the body were transferred to liquid of specific gravity of 1-2, what fraction of its volume would then project above the surface?

No. 39. — Elementary Mechanics. — For Class D. Time allowed: Three hours. [All answers should be illustrated, where possible, with diagrams.] 1. Describe how velocity is measured (i.) when uniform and (ii.) when variable. If the velocity of a body is 30 when measured in feet and seconds, what is it when measured in miles and hours ? 2. Describe briefly how you would explain and illustrate to a class (i.) the formula s=ut + iat 2 , where a represents the acceleration ; (ii.) the distinction between the weight of a body and its mass. 3. If a body falling freely describes a distance of 20 yards in one particular period of a third of a second, what was the velocity at the beginning of the period ? 4. State the principle of the parallelogram of forces, and describe any modifications of it that are sometimes useful, showing what uses may be made of them.