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c. Explain the apparent irregularity of the preformative ; in Toda, and
(. 8). do przo (ver. 9). Explain the 3 in this word, and give the root. XIII. Give an account of Dr. Wall's theory of the vocalization of the Hebrew Bible,
Examiners. HUMPHREY LLOYD, D. D., Vice-Provost. John Lewis MOORE, D. D. THOMAS LUBY, D. D. ANDREW SEARLE HART, LL. D. CHARLES GRAVES, D. D. THOMAS STACK, M. A., Regius Professor of Greek. JOSEPH A. GALBRAITH, M. A., Professor of Experimental Philosophy. John K. INGRAM, LL. D., Professor of English Literature.
DIFFERENTIAL AND INTEGRAL CALCULUS.
1. Find a maximum or minimum value for y in the equation
(y-1) (x+y + 1)2 = (x+2y+3)
ex-e-2 - 23 2. Find the value of
when a = o. tan 3 - 30
3. Find whether (1+x)= is capable of a maximum or minimum value. 4. Change the independent variable from a to 0 in the equation
da where x = tan 0.
5. Find the length of the arc of a parabola cut off by the focal ordinate.
6. Compute the area and length of the cycloid whose equation is
= a cos -1
(-->) - v 'zay=y.
7. Find the area of the surface and the volume of the solid generated by the revolution of this cycloid about its base. 8. Find the number and nature of double points on the curve
24 – 40x3 + 4a x2 – 5242 +263y – 24 – 64=0. 9. Find the form of the curve 8x +y t = at, and the length of its circumference.
12. Seek a point from which if perpendiculars be drawn to the three sides of a triangle, their fourth proportional shall be a maximum or minimum, and explain the result.
1. What are the conditions that the equation
6x3 + Axạy + Bxy2 +7ax? + Cxy + Dy2 + a3 = 0 should represent three right lines ?—a being given, and A, B, C, D to be determined.
2. What additional condition is necessary that these lines should pass through the same point ? 3. Find the magnitude and position of the axes of the ellipse
5x2 + 2xy + 5y2 + 4ax — 28ay + 20a2=0, ist, if the axes be rectangular;
2nd, if they make an angle of 60°. 4. Find the length of the chord which is normal to a given ellipse at a given point.
5. Find the locus of the poles with respect to a given conic of the tangents to another conic.
6. Find the locus of the centres of conics touching the three sides of a given triangle and the circumscribing circle.
7. If one focus of a conic, inscribed in a given triangle, lies on the circumference of the inscribed circle, find the locus of the other focus.
8. Find the locus of the intersection of normals at the extremities of a chord which passes through a given point.
9. Find the envelope of chords of a conic which subtend a given angle at a given point on the curve. 10. Deduce the equation of the tangent in the form,
Y from the equation of the line joining two points on the conic
✓la + mß +vny = 0. 11. Find the envelope of ellipses the position of whose axes are given, and
ist, their sum; or
ALGEBRA AND TRIGONOMETRY.
27 1. Prove the equations cos + COS
+ cos — 7
3 7 57
571 sin sin
14 14 21 2. If A be the area of a spherical triangle, deduce the value of tan a from the values of the sines and cosines of the half angles in terms of the sides. 3. Find the number of real roots of the equation
201 - 5.02 – *+7= 0, and approximate to their values as far as the first decimal place. 4. If three roots of the equation
*c* + poc3 + 0x2 + rx+8=0 be proportional, prove
212 + 739 + 9826 + ps233 + = 0. 5. Find the condition that
* +px3 + 9x2 +rx+8 = 0 shall have two equal roots.
6. What additional condition is necessary that three of its roots may be equal ?
1. State the formula of interpolation which is to be used for equal interyals.
2. Apply it to the following case, proceeding as far as second differ rences. The distance of the Moon's centre from a star is as follows :
At Noon, 34° 6' 18"
IIIh. 35° 28' 13"
Vih. 36° 51' 13" Calculate the distance at 1h 31m 51%.
3. The declinations of the Sun at noon, taken from the Nautical Almanac, are as follows:
23° 26' 55" 20,
23° 27' 25":5 21,
23° 27' 31":3 22,
23° 27' 12".2 Find the time at which it reaches its greatest declination.
4. The front of a house in latitude 50°44' N. makes an angle of 25° with the meridian from the N. towards the W. At what o'clock on midwinter-day will the Sun commence to shine on the rooms on that side ?
5. Assuming the periodic time of Jupiter to be 4332 days, and the interval from eastern to western quadratures to be 174 days; calculate his annual parallax, and distance from the Sun, as compared with that of the Earth.
6. For a given latitude, find the declination of a heavenly body which in its diurnal motion will pass from one given parallel of altitude to another, in the shortest possible time.
7. Explain the method of finding the latitude at sea by observing a double altitude of the Sun. State the most advantageous formulæ for the solution of each triangle ; also the mode of allowing for the ship's motion.
8. In order to calculate the base of the first triangle, it is usual to consider it isosceles, each side being the mean value of the two codeclinations. Show that the error on the base is expressed as follows, in terms of the difference of declinations :
cos } C (diff.")2 (Error)" = }
206265' C being the vertical angle, and c the base.
9. A transit instrument is placed approximately the meridian; in order to discover the deviation in seconds E. or W. of the true meridian, an observation is made of the time of passage of an equatoreal star, and also of that of the upper transit of a circumpolar star; then if
A (seconds) = diff. of R. A. - diff. of times,
cos d cos d' (Deviation)" = AX
sin (d' - d) cos lat. 10. Another way of obtaining the deviation is to observe the interval between an upper and lower transit of a circumpolar star, e. g. Polaris (Dec. = 88° 34'), and subtracting 12 hours; show that for the latitude of Dublin,
Remainder in seconds (Deviation)" =
1. In a system of bodies, prove that the sum of the products of the masses into the squares of their distances from a given point will be a minimum, when the point is the centre of gravity of the system.
2. Let three forces be applied perpendicularly to the middle points of the sides of a triangle, and in its plane; prove that they will be in equilibrio, when they are proportional to the sides of the triangle.
3. Two heavy spheres rest on two inclined planes, and against each other; find their position of equilibrium.
4. A heavy straight rod moves in a vertical plane round a hinge at the lowest point, and is supported by a weight connected with its upper ex