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2. A person sowed a bushel of wheat, and the next year he sowed again the produce of that bushel, and so on until 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
587 value of
1943 2. Show in the above example that the approximating fractions must be alternately greater and less than the true value.
230 3. How can the fraction be divided into two others, whose
77 denominations are 7 and 11 ?
MECHANICS AND ASTRONOMY.
Section 1. 1. Define the unit of work, and show that if a pressure of m lbs. be exerted through a space of n ft., the number of units of work done is represented by m X n.
2. How many tons of coals can be raised in 24 hours, from a a depth of 20 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 fraction of 200 lbs., 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 economised 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 feet 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 a 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 ibs. ?
3. A block of cast iron weighing 100 lbs. rests upon a plank of oak, inclined at 24 deg. 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 deg. ?
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. What are the astronomical causes of variation of temperature ?
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 10 h. 11 min. 43 sec. apparent time. The apparent Greenwich time of the immersion was, by the Nautical Almanac, 8 h. 13 min. 56 sec.
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 deg. travelling due east in latitude 45 deg. ?
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.
EXAMINATION PAPERS PROPOSED AT THE CHESTER DIOCESAN TRAINING COLLEGE.
Section 1. Give dates for the following events :1. The call of Abraham and the dedication of Solomon's temple.
2. The Exodus and the decree of Cyrus for the rebuilding of the temple.
3. The day of Pentecost and the captivity of St. Paul at Rome.
Section 3. 1. Draw a plan of the temple as it existed in the time of our Lord.
2. Draw a map of Jerusalem.
2. The daily morning and evening sacrifices; and point out their typical character.
Section 5. What prophecies have reference to the time of the advent of the Messiah, and how have they been fulfilled ?
Section 6. What events in the lives of,
2nd. Peter, afford subject for practical instruction ?
Section 7. State shortly the events which occurred between our Lord's agony in the garden and his ascension.
Section 9. 1. Write down the names applied in Scripture to our Saviour.
2. What offices are intimated under the name Christ, and in what sense have those offices been fulfilled ?
3. Show that the Messiah was expected by the Jews as a temporal ruler.
CHURCH HISTORY.-(PALMER.) EVIDENCES OF CHRISTIANITY.-(PORTEUS.)
(One question only to be answered in each section.)
Section 1. 1. Give some account of the persecutions of the Christians of the second century.
2. Who were the most illustrious of the fathers of the two first centuries !
Section 2. 1. What authority of Scripture and of antiquity is there for the use,
in the services of the Church, of a language " understanded of the people ?"
2. When, and upon what pretence, were Latin services introduced in the Churches of northern Europe ?
Section 3. 1. Give some account of the introduction of image worship, and of the councils by which it was respectively supported and condemned.
2. When, and under what circumstances, did the division of the Eastern and Western Churches take place?
Section 4. 1. What testimony is borne to the piety of King Alfred ? 2. Give some account of Robert Grosteste, bishop of Lincoln.
Section 5. 1. Relate concisely the principal events in the public life of Martin Luther, giving as many of the dates as you can.
2. How long did the Popish party remain in communion with the Church during the reign of Elizabeth, and under what circumstances did they separate themselves ?
Section 6. By what argument is it proved
1. That the books of the New Testament were written by the persons to whom they are ascribed ?
2. That in the facts they relate they could not be deceived themselves, and could have no inducement to deceive others ?