6. What is the geometric signification of the co-efficients of an

equation y = ax + b, 1/2 + 2 = 1

a

1?

7. What is the length of the perpendiculars from the point (xy) on the straight line ay + bx + c =o? Prove it.

Find the tangent of the angle between two right lines?

8.

10. If L—L'=o is the bisector of the angle between the two lines, what is the form of the equation?

11. Hence prove that the bisectors of the angles of a triangle meet in a point.

12. Find the equation of a straight line, in terms of the perpendiculars on it, from the origin and the angle it makes with the axis?

13. If a a= o, B

=

0, y =

o, be the equations of the three sides of a triangle, what is the equation of the perpendicular from the angle on the opposite side?

14. Hence the perpendiculars from the angles on the opposite sides meet in a point?

15. The equations of the bisectors of the sides are 16. Hence they meet in a point?

?

17. What is the equation of the line passing through the centre of a circumscribing circle?

18. The perpendiculars at the middle points of the sides meet in a point?

19. If a, ß, y, be the equations of the sides of a triangle, how do you represent the equations of lines passing through the angles and through a point?

20. What are the equations of the lines passing through the intersection of each pair with the opposite sides?

21. The intersection of these lines with the third side lie on the straight line la+mß+ny=o?

23. Find the equation of the tangent to the circle

y2 + x2 + ay + bx + c = o

24. Find the equation of the chord of contact.

25. Find the radius and co-ordinates of the centre of the circle

26. Prove that the length of a tangent to a circle from a point without it is y 2 + x2 + ay + bx + c =

= 0.

27. Hence find the equation of the line from which equal tangents can be drawn to two circles.

28. The radical axes of three circles pass through the same point? 29. When does the general equation of the second order represent straight lines?

30. In the general equation of the second order ay2 + bx y + c x2 + dy + ex+f=o find the value of y in terms of x.

31. What relation exists among the co-efficients, when the roots of the quantity under the radical are real and equal, real and unequal, and imaginary?

32. When (2-4 a c) is negative, the locus is an ellipse?

33. What varieties does this admit of?

34. When (2-4 a c) is positive, the locus is an hyperbola ? 35. What varieties does this admit of?

36. When (62 — 4 a c) o the locus is a parabola?

37. What varieties does this admit of?

2

38. Show that the equation a y2+ bxy + c x2 = o represents two straight lines.

39. Prove generally that the homogeneous equation of the nth order represents straight lines.

JOSEPH PATTON.

PAPER.

1.

Demonstrate the formula for finding the time by an altitude of a heavenly body, and explain the method of determining the longitude by a chronometer.

2. Length of to-day at Bombay, and times of sunrise and sunset? -prove the formula.

3. How is the index error of a sextant determined? Explain the corrections of an altitude for parallax and refraction.

4. Determination of the latitude by observations with a transit instrument in the prime vertical?

5. Formula for the angles on a horizontal dial?

6. Prove that when the sun's declination is greater than the latitude of a place on the same side of the equator, the shadow of an object perpendicular to the horizon goes back during a portion of the day.

7. The base of a system of logarithms is 6, what is the modulus? Demonstrate the formula.

8. Application of logarithms to the subject of annuities?

9. A and B are two fixed points: draw through B any line, and let fall on it a perpendicular from A, AP; produce AP so that the rectangle AP, AQ may be constant ; to find the locus of the point Q? 10. Given the vertical angle of a triangle and the sum of the reciprocals of the sides, the base passes through a fixed point?

11. Given base and ratio of sides in a triangle, to find the locus of the vertex?

12. Polar equations to the conic sections?

13. The parabola is an ellipse whose second focus is infinitely distant?

14. Theory of asymptotes? Show that the hyperbola has an asymptote, and that the ellipse and parabola have not.

15. Locus of the intersection of the focal radius in an ellipse with the radius of the circle on the major diameter ?

16. Given the base and the sum of the sides of a triangle, to find the locus of the point of intersection of lines from the angles bisecting the opposite sides?

17. What is the principal parameter of the parabola whose equation is y2-2xy + x2-3y=o?

H. GREEN.

2ND NORMAL SCHOLARS.

DIFFERENTIAL CALCULUS AND MECHANICS.

(Viva Voce.)

1. Give some instances in which quantities vanish whilst their ratio continues to exist.

2. What is meant by one quantity being a function of another? 3. What is the object of the differential calculus ?

4. What geometrical problem gave rise to the differential calculus?

5. To draw a tangent to a curve is the same as finding a differential co-efficient?

6. If u be a function of y, and y a function of x, then

7. Show that dn (u v) may be written by (d+d')" u v, and state the meaning to be attached to d and d'.

8. On what principle does the proof of Taylor's theorem depend? Prove the theorem.

9. Maclaurin's theorem may be derived at once from Taylor's? 10. In the expansion of the sine x in terms of x there can be no even powers?

11. In the expansion of the cosine x there can be no odd powers?

12. Prove the exponential values of cos x and sin x.

13. From these show how to find the factors of

15.

What is meant by vanishing fractions?

16. How is the differential calculus applied to find their value? 17. What is the condition of a function being a maximum or minimum?

18. When is it a maximum ?-when a minimum ?

19. How is the calculus applied to determine asymptotes?

20. What is meant by the osculating circle?

21. How is this expressed analytically?

22. Find the radius of curvature at any point.

23. Hence show the conditions for a point of inflexion.

24. Show geometrically that the normals to a curve are tangents to the evolute.

25. Also that an arc of the evolute is equal to the difference of the radii of curvature at its extremities.

26. If an ellipse and circle intersect in four points, the lines joining the points of intersection make equal angles with the axis?

27. If the circle touch the ellipse, the line joining the two points of intersection and their common tangent make equal angles with the axes?

28. Hence a very simple method of describing the osculating circle of the ellipse?

31. In a force acting on a material point, there are three things. to be considered?

32. Explain how forces can be represented by lines.

33. Give a geometric construction to find the resultant of a number of forces acting on a material point in the same plane.

34. What are the equations of equilibrium of a point acted on by any number of forces?

35. Granting that the resultant of two equal forces is the diagonal of the parallelogram, show that it is true for any two perpendicular forces: hence prove the general principle.

36.

Let PP' be any two forces and R their resultant, and let p, p', and r be the perpendiculars on their directions from any point, prove that Pp P'p'

= =

Rr.

37. Deduce, hence, the condition of the equilibrium of the lever, namely, Pp

P'p'.

38. What is the condition of equilibrium in the funicular polygon?

39. In a suspension-bridge, how do the tensions of the sides vary?

40. Where is the tension least?

41.

Find the relation between the power and the weight on the inclined plane. Find the true weight by means of a false balance. 42. Prove the equations for finding the centre of gravity of parallel forces X = 2(Px) Y= Σ (Py) and Z = Σ (Ρ2)

ΣΡ

ΣΡ

ΣΡ

43. How are these applied to find the centre of gravity of a solid, a surface, a line, and a triangle?

44. Stable and unstable equilibrium?

Paper.

JOSEPH PATTON.

1. Explain the doctrine of limits. Give illustrations.

2. In the general form of the development of ƒ (x + h), viz. f(x)+ Ah+Bh2 + &c. show that none of the exponents of h can be fractional.

3. The nature of the so-called "failing cases" of Taylor's theorem?