II.-Trigonometry. 1. SHOW that sin. (a + b) sin. a cos. b + cos. a sin. b. 2. Find the sines of 30°, 15°, and 18°. 3. Prove the formula, sin. (a + b) sin. (a — b)— sin. 2a —sin.2 b. -1 -1 4. Show that tan a + tan −1 b = tan ' -1 ( a +0%). 6. Given the sides a and b and the angle C in a triangle, find the perpendiculars let fall from the angles on the opposite sides. 7. Explain the method of proportional parts in finding logarithms. DIFFERENTIAL CALCULUS. 1. Define a differential coefficient, and from your definition du determine in the following cases. dx 2 u =√x2 + 1, u = sin. −1 x, u = log. sin. x. 2. Differentiate (sin. x) *, and ex cos. x. 3. Show that in general the development of ƒ (x + h) can only contain positive and integral powers of h. 4. Prove the rule for finding the value of a vanishing fraction in all its cases. 5. Prove the rule for finding maxima and minima. 6. Inscribe the greatest isosceles triangle in a given circle. 9. Explain generally the theory of contact of different orders, and apply this theory to determine the radius of curvature. STATICS. 1. Determine the resultant of two parallel forces acting in opposite directions on a rigid line. 2. Find the centre of gravity of a triangular pyramid. 3. Find the ratio of P to W in the common screw. 4. If a sphere A of given radius be placed upon the top of another sphere B which is fixed, determine when the equilibrium will be stable, and when unstable. 5. A B, B C, C D, D A, and A C, are five equal uniform rods, each weighing W lbs., jointed together and suspended from B; find the thrusts on A C, and the strains upon A D and D C. III.-Euclid. A B 1. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles. 2. If a right line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line. 3. In every triangle, the square of the side subtending either of the acute angles, is less than the squares of the sides containing that angle, by twice the rectangle contained by either of those sides, and the right line intercepted between the perpendicular let fall upon it from the opposite angle, and the acute angle. 4. Equal right lines in a circle are equally distant from the centre; and those which are equally distant from the centre are equal to each other. 5. In a circle, the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle. 6. To describe an isosceles triangle, having each of the angles at the base double of the third angle. A A |