MOTION IN A RESISTING MEDIUM. 1739. Find the time in which a body of given weight falling from rest through the air will acquire a velocity v; assuming that the resistance varies as the square of the velocity. 1740. Find the space through which the body will have fallen when it has acquired the velocity v. 1741. If a particle be projected in a medium the resistance of which varies as the velocity, find the space described in the time t, supposing no other forces to act. 1742. A body is projected vertically upwards with a velocity V; find the height to which it will rise, the resistance of the air varying as the square of the velocity. 1743.. If in 1741 the resistance vary as the square of the velocity, find the space described in the time t by a body whose weight is w. 1744. A body is projected with a velocity V obliquely into the air at a small angle of elevation a; show that, if the vertical resistance of the air be neglected, the range on a horizontal plane is where w is the weight of the body, and k the resistance due to a unit of velocity. 1. Deduction. If two sides of a triangle be bisected, show (from the First Book of Euclid) that the line joining the points of bisection is parallel to the third side, and equal to half that side; and thence show that if all the sides of a quadrilateral figure be bisected, and the adjacent points of bisection be joined, the figure so formed will be a parallelogram equal to half the given quadrilateral. 2. In every triangle the square of the side opposite any of the acute angles is less than the squares of the sides containing that angle, by, &c. 3. If two straight lines within a circle cut one another, the rectangle contained by the segments, &c. 4. Equal triangles which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional; and triangles which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal. 5. The circumferences of circles are to one another as their diameters. 6. If two straight lines meeting one another be parallel to two other straight lines which meet one another, but are not in the same plane with the first two; the plane which passes through them is parallel to the plane passing through the other. COORDINATE GEOMETRY. 7. Investigate the relation between a and a', so that the lines y = ax + ẞ, y = ax+ B', may be perpendicular to one another. The line yax+b passes through the point (1,-2), and is perpendicular to the line 5y - 10x + 12 find the values of a and b. = 0; 8. Construct the circle denoted by the equation x2 + y2-6x+10y - 15 = 0; and find the position of that diameter of it which passes through the origin of coordinates. A. II. ARITHMETIC AND ALGEBRA. 1. Define the terms "fraction," "power," "root," "index," "logarithm" and "modulus." Explain also the reasoning by which it is shown that a-(b-c) = a − b + c. 2. A person owes £800 bearing interest at 5 per cent. per annum. At the end of each year he pays £120 for interest and in part payment of the principal. Find the amount of his debt at the end of the second year. (a+b)2 - (c + d) 2 to its most simple form; and find 3. Reduce (a+c)2 - (b + d)2 a+b+c the value of 4. Solve the following equations: 5x+6= 2 1 ... == x + (1); 6 (x − y) (a2x2 + b3y3) = a3 (x3 — y3), xy = c3 ......... (3). 5. A and B agree to pay their expenses for a certain time in the proportion of the numbers 4 and 7. At the end of this period it was found that A had paid the sum of £102, and B £73. What has the one to pay and the other to receive in order to settle the account? 6. The equation x3 — x2 - 33x - 63 = 0 has two equal roots: find them by means of the derived equation; find also the third root. 7. Define a geometrical progression, and show that if each term be subtracted from the preceding, the successive differences constitute also a geometrical progression. Sum the latter series to n terms, when the first term of the original series is 2 and ratio 1 3 8. Investigate the formula for the number of shot in a square pile; and show that if the number in a square pile be to the number in a triangular one of the same number of courses as p to q, then the number of courses in each is |