its distance from the points A B C; or show how these may be determined by calculation. Section 6. 1. Define the logarithm of a number N to the base a; and show that 2. Expand a*. log M=log M - log N. 10 10 3. Investigate an expression for determining the logarithm of a number to any given base in a converging series. Section 7. 1. Investigate an expression for the area of a quadrilateral figure inscribed in a circle. 2. Expand Sin mx in terms of Sin x. 3. Determine either angle of a spherical triangle in terms of its sides. Section 8. 1. Investigate a rule for determining the area of a trapezoid. 2. What is the area of a circular plot of ground whose diameter is 27 chains? 3. The side of an octagon is 10 feet, what is its area? 4. A ring is generated by the revolution of an equilateral triangle whose side is three inches about an axis parallel to one of its sides and distant 6 inches from it. What is the solidity of the ring and its surface. Section 9. 1. One side of a rectangular field is double the other; the field measures 20A. OR. 19p. what are its sides? 2. Given the several offsets, 15, 25, 40, 10, 60, 30, 25, 8, 18, 9, 8, 4, 6, 0, taken at one chain's length. Required the area. 3. Required the plan and content of a four-sided field contained by straight lines according to the following field-book 1. At what price per head must a farmer purchase a flock of 100 sheep, that, expending 107. in feeding them, and losing 9, he may be able to sell the remainder at 27. each and gain 207. ? 2. I turn over the pages of a book by fours, and find three odd ones. I then turn them over by fives, and find two odd ones. The last time I do not turn them over so often by twenty times as I did the first. How many pages were there? 3. A passenger train and a luggage train, the one travelling at 10 miles per hour less speed than the other, set out at the same time, the one from London and the other from Carlisle, 210 miles apart, and pass one another at a certain station on the road. The passenger train sets out from Carlisle to return, two hours after the luggage train sets out to return from London; and it is observed that they pass one another at the same station. At what rate do they travel, and how far from London is the station? Section 7. 1. The first term of an arithmetical progression is -7, the number of terms 8, and the sum 28. What is the common difference? 2. A person sowed a bushel of wheat, and the next year he sowed again the whole produce of that bushel, and so on until at the end of the third year he had a bushels. How many grains of wheat must each grain of seed have yielded, supposing it to have yielded the same number every year? 3. What is the present value of an annuity of £a, which increases in a geometrical progression, whose ratio is r for n years, interest being assumed at p per cent. per annum. Section 8. 1. Approximate by the method of continued fractions to the value of 587 1943 2. Show in the above example that the approximating fractions must be alternately greater and less than the true value. 3. How can the fraction nators are 7 and 11. 230 be divided into two others, whose denomi MECHANICS AND ASTRONOMY. [(One Question only is to be answered in each Section.) "1. Define the unit of work, and show that if a pressure of mlbs. be exerted through a space of n. the number of units of work done is represented by mxn. 2. How many tons of coals can be raised in 24 hours, from a depth of 90 fathoms, by a winding engine of eight-horse power? 3. In how long a time would a five-horse power engine empty a shaft 8 feet in diameter and 200 fathoms deep, which is full of water, and what would be the expense in coals if the engine did 30 millions duty ? Section 2. 1. How many horses, each exerting a traction of 200lbs., would be required to draw a waggon weighing, with its load, 5 tons, up a hill whose inclination is 1 in 18; the traction on the level road being estimated at 1-20th of the load? 2. A shaft 256 feet in depth is to be pumped dry by three men working in succession; to what depths must they respectively sink the surface, that each may do an equal share of the work? 3. On what principle is the power of an engine economized by working it expansively? Section 3. 1. The section of a stream is 5 feet by 7, its mean velocity is 3 feet per second, and there is a fall upon it of 11 feet, working a wheel whose modulus is 6: how many quarters of corn will it grind in 12 hours, allowing a bushel per hour to each horse-power? 2. A stone is let fall from the top of a tower 100 feet high, and at the same instant another is projected upwards from its base with a velocity of 20 ft. per second, where will they pass one another? 3. A cubical block of stone rests upon a railway truck: what must be its dimensions (the weight per cubic foot being given), that it may just be overturned when the train, travelling with a given velocity, is suddenly stopped? Section 4. 1. An iron bar 4 feet long and weighing 15 lbs. is supported at its extremities in a horizontal position, and a weight of 28 lbs. is suspended at I foot from one end: what is the pressure upon each point of support? 2. What lading will a rectangular barge carry whose length is 30 feet, breadth 6 feet, and depth 4 feet, each square foot of the iron weighing 9 lbs. ? 3. A block of cast iron weighing 100 lbs. rests upon a plank of oak, inclined at 24° to the horizon: what pressure acting parallel to the plane will draw it up it, and what down it; and what is the direction and amount of the least pressure which will draw it up and down; the limiting angle of resistance between iron and oak being 32°. Section 5. 1. How is it known that the earth is not a plane, and how is it known that it is a sphere? Give one reason, and the simplest, in each case. 2. Describe the apparent motions and the variations in brightness of one of the inferior planets. 3. Show that the apparent motions of the planets would be the same as we now observe them to be, if, revolving round the sun, they also revolved with him round the earth? Section 6. 1. Show that the latitude of a place is equal to the apparent elevation of the pole at that place. 2. Explain the seasons. of temperature? What are the astronomical causes of variation 3. The moon revolves round the earth in 27 3 days, and the period between one new moon and another is 29.5 days. How is one of these numbers deduced from the other? Section 7. 1. An immersion of one of Jupiter's satellites was observed at 10h 11′ 43′′ apparent time. The apparent Greenwich time of the immersion was, by the Nautical Almanac, 8h 13′ 56′′. What is the longitude of the place of observation. 2. Why do not eclipses return every month? Why are eclipses of the moon visible at all places of the earth's surface where the moon is visible, and eclipses of the sun only at certain places where he may be seen? 3. How often would the same eclipses return if there were no regression of the moon's nodes? Section 8. 1. How many geographical miles would a man travel who changed his longitude 10° travelling due east in latitude 45°? 2. State Kepler's law of the equal description of areas, and prove it on mechanical principles. 3. Investigate a formula for determining the hour angle from an observed altitude of the sun; and explain how the true time is determined from the hour angle. |