4. Define the evolute of a curve. Investigate the property on which it depends; find the evolute to the cycloid.

5. Determine the nature of the curve whose equation is y3 + 23. ax2 = 0, find the maximum ordinate, and point of inflexion. Trace and find the area of the curve whose equation is

=

6. If in the radius vector SP of a parabola, (the vertex of which is A, and Sy the perpendicular from the focus S upon the tangent at P) a point Q be taken, such that SA Sy SQ SP, find the equation to the curve which is the locus of Q; trace the curve and shew that the areas of the curve and parabola between the vertex and the latus rectum of the parabola are as 3: 4.

7. Shew how to find the length of a curve referred (1) to rectangular co-ordinates, (2) to polar co-ordinates. Prove that the length of the curve whose equation is x3+yR = a intercepted between the axes of

= a}

8. Find the volume of the solid generated by the revolution, about the axis of x, of the lemniscata the equation of which is

(x2 + y2)2 = a2 (x2 — y2)

SECOND CLASS.

Afternoon Paper.

1. State the steps in the reasoning by which it is shewn that f(x + h) admits of development in a series proceeding by ascending positive and integral powers of h.

2. If (u) be a function of y, y a function of

du du dy
X, =
dx dy dx

this proposition to differentiate by substitution the function. Required the differentials

Employ

hyp. log. ✔a2 — x2, xmx and 2√√—1 (ex√√/−1 +ɛ ̄ï√—1)

a2 + x2

3. Define a multiple point, and shew from the definition that if dy

dz

be obtained from the equation to the curve made free of radicals, the coordinates of the multiple point will make it assume the form

THEORY OF EQUATIONS AND CONIC SECTIONS.

xxi

Take as an example the curve a2y2 = a2 (x2 — y2) and determine the direction of its branches at the multiple point.

4. A curve is convex or concave to the axis of x, according as has, or has not, the same sign as the ordinate.

day

da

Determine the minimum value of (x — a)m m being odd.

5. Find the differential expression for the radius of curvature, and shew that it agrees with Newton's.

Ify and a be functions of a third variable 0, the expression for the radius of curvature is

determine what this expression becomes when is the arc of the curve.

6. Trace the curves defined by the equations

7. Investigate the differential expression for a surface of revolution; and find the surface generated by the revolution of the lemniscata, the polar equation of which is r2 = a2 cos 20.

8. Find the locus of the intersection of the perpendicular, drawn from the vertex, and tangent to any point of a parabola. Trace the curve and find the area between the curve and its asymptote.

* Sx (a2 + x2)n Sq sin m9 cos no.

9. Integrate Sr (a + br3) * Sx (a2 +

1

of

Make the integral of Sa (a2+22), depend on that of

x

1

10. Obtain the integral of Sa√ a + bx + call

THIRD CLASS.

1

x(a2 +x2)n−1

THEORY OF EQUATIONS AND CONIC SECTIONS.

Afternoon Paper.

1. Shew how to transform an equation into one which shall want the second or third term; under what circumstances can both be made to disappear at one operation?

Form an equation of six dimensions having the co-efficients of the 2nd and 3rd term so related that they can both be taken away at one operation. 2. The limiting equation must always have as many possible roots as the original wanting one.

Hence prove that if m consecutive terms be wanting in an equation, it cannot have more than (n-2m.) possible roots. How many possible roots can the equation an · ax2 + b =o have.

3. Give Cardan's method for the solution of a cubic equation.

Shew that it fails when all the roots are real, and succeeds when two roots are imaginary, or when all real but two equal.

4. If several roots of an equation lie between two consecutive integers, how may Sturm's Theorem be applied to find an approximation to each ?

Find by this method an approximate value of a root of the equation x3-x2- 5 = = o. Correct to three places of decimals.

5. Explain Newton's method of approximating to the roots of an equation, and shew when it may safely be applied.

Obtain an approximate value of a root of a3+4 x2 rect to two places of decimals.

1 = 0. Cor

6. Define the asymptotes of an hyperbola. If any straight line Qq perpendicular to either axis of an hyperbola meet the asymptotes in Q and y and the curve in P the rectangle Q P. Pq is invariable.

=

7. In the Ellipse the sum of the squares of the conjugate diameters is constant (C P2 + C D2 : A C2+ B C2.) If the normals at P and D intersect in K shew that K C is perpendicular to P D. 8. If any chord AP through the vertex of an hyperbola be divided in A C2: B C2, and Q M be drawn perpendicular to the foot of the ordinate M P shew that Q O at right angles to Q M cuts the transverse axis in the same ratio.

Q so that A Q: QP

FOURTH CLASS.

EUCLID AND ALGEBRA.

Afternoon Paper.

1. Upon stretching two chains, AC, BD, across a field ABCD, I find that BD and AC make equal angles with DC, and that AC makes the same angle with AD, that BD does with BC. Hence prove that AB is parallel to CD.

2. Determine the regular polygons which by juxta-position may fill space about a point, all of them being situated in the same plane. What advantages arise from the honeycomb consisting of hexagonal cells. 3. ABC is an equilateral triangle; E, any point in AC; in BC produced take CD CA, CF CE; AF, DE, intersect in H. HC

=

EC

AC

AC+ EC

4. If three clocks were regulated to go in the following manner; being set at 12 o'clock at noon on the first of January 1852; the first to keep the exact time, the second to gain a minute, and the third to lose a minute per day; what day, month and year would they meet again at the same hour.

5. Shew how to transform a number from one scale of notation to another. Having given 16:34 in the octenary scale and 0545 in the senary, find their product in the undenary scale. Find the area of the rectangle 4 yards, 1 foot, 2 inches long, 3 yards, 2 feet, 4 inches wide. 6. Find the sum of the series

2) + ..

mn+ (m1) (n − 1) + (m − 2) (n − 2) +

Hence find the number of balls in an incomplete rectangular pile, of 22 courses, which contains 68 balls in the length and 44 in the breadth of the bottom row.

7. Expand a in a series ascending by powers of x.

Shew that

1 1 1.2 123

1+1+ +

+

1 1.2.3.4

+ &c. to infinity is con

vergent, and that its limit cannot exceed 3.

8. An urn contains 20 balls, 4 of which are white,

If a person draw 5 at a venture, find

1. the probability of drawing only one white ball.

2. the probability of drawing at least one white ball.

9. If the terms of the expansion (a + b)m be multiplied respec

ber, find the sum of the resulting series.

10. Find the present value of a scholarship of Rs. 40 per month (payable monthly), the enjoyment of which is to commence 5 weeks from this date, and to continue for 12 months, at 5 per cent. simple interest.

11. A railway train after travelling for one hour meets with an accident, which delays it one hour, after which it proceeds at ths of its former rate and arrives at the terminus 3 hours behind time; had the

accident occurred 50 miles further on, the train would have arrived one hour and twenty minutes sooner; required the length of the line.

FIRST CLASS.

OPTICS.

Morning Paper.

1. Define a pencil of rays, converging rays, diverging rays, and the focus of a pencil of rays.

If diverging or converging rays be reflected at a plane surface, the foci of the incident and reflected rays are on contrary sides of the reflector, and equally distant from it.

Why does the common looking glass give more than one image at a point?

2. Find the geometrical focus and aberration for a pencil of rays converging to a given point between the centre and principal focus of a convex mirror, and shew that, whether the rays be divergent or convergent, the aberration is towards the mirror.

3. A small pencil of rays is incident obliquely on a concave refracting surface; find the positions of the focal lines, and shew for what values of u the primary focus is further from the surface than the secondary, drawing the requisite figures.

4. Find the deviation of a ray after two successive reflections at plane mirrors inclined to each other at a given angle, the course of the ray lying in a plane perpendicular to their line of intersection.

What must be the first angle of incidence that at a third reflection the course of the ray may be exactly reversed?

5. If a ray of light passes through a glass prism shew that it is bent towards the thicker part of the prism, and that the deviation (μ—1)r when the reflecting angle r, and the angle of incidence are both small. Hence deduce the position of the principal focus of a double convex lens. Why is called the power of the lens.

6. Find the principal focus of a refracting sphere. How may a sphere be used as a microscope?

7. What is the dispersive power of a transparent medium, and how is it measured? What is a table of dispersive powers? Give a short account of irrationality of dispersion, and secondary and tertiary spectra.