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SENIOR FRESHMEN.

Mathematics

A.

DR. STUBBS.

1. Differentiate

sin -1

2cx + b

✓b2-4ac

and log {2cx + b + c√/c√/ a + bx + cx2 }

and find the ratio of the differentials in its simplest form.

2. The equation of the cissoid is y2 (2r − x) — x3 = 0; find the equation of the tangent, and show that there will be an asymptote to the curve. 3. Prove by the method of projections that the radius of a circle cirb' b" b"" cumscribing a triangle inscribed in a conic is bb"b" being the

semidiameters parallel to the sides of the triangle.

ab

4. Prove (1) that confocal conics cut at right angles, and (2) recipro

cate this theorem.

5. The equation x+6x3 + 6x2- 9x-4=0 may be reduced to a quadratic, and solved by putting x2 + px=y.

6. Find the roots of the cubic equation which gives tan x in the equation_m tan x = tan (y - x) when x+y=30°.

MR. TOWNSEND.

7. Prove the values for the perpendiculars of a spherical triangle in terms of the sides.

8. Prove the formula for the area of a spherical triangle in terms of the sides.

9. Tangents to a central conic at right angles to each other intersect on a circle concentric with the conic; transform this property by reciprocation from an arbitrary point.

10. Tangents to a parabola at any constant angle to each other intersect on a conic having a common focus and directrix with the parabola ; transform this property by projection from an arbitrary centre.

11. Eliminate by differentiation the constants a and b from the equation in two variables (x − a)2 + (y — b)2 = r2.

12. Investigate the equation of the evolute of the curve whose equation in rectangular co-ordinates is xy = a2.

t

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14. Deduce the expression for the radius of curvature of a plane curve in polar co-ordinates; and apply it to find the radius of curvature of the curve r = aemo.

15. Find the co-ordinates of the centre of curvature, and the equation of the evolute of a parabola.

16. Find the equation of the circle osculating a parabola.

17. A cord PP' of an ellipse is drawn perpendicular to the axis major AA'; find the locus of the intersection of AP′ and A'P.

18. If one of the arcs of great circles which join the middle points of the sides of a spherical triangle be a quadrant, show that the other two are also quadrants.

B.

DR. STUBBS.

1. The sides of a triangle are 3, 4, and 5, and its inscribed circle has been removed; find the distances of the centre of gravity of the remainder from the first two sides.

2. An uniform rod rests with one end on a smooth hemispherical bowl, and the other against a fixed vertical rod passing through the centre of the hemisphere; find the position of equilibrium, and the pressures against the bowl and the upright rod.

3. A door whose hinges are in a line inclined to the vertical at an angle a is raised through an angle ẞ from its position of equilibrium ; what force perpendicular to the plane of the door will keep it in this position and calculate the strain upon the hinges.

4. A beam rests on two inclined planes; find the position of equilibrium when the centre of gravity divides it in a given ratio.

MR. TOWNSEND.

5. Investigate, by any method, the condition that a system of any number of forces in space should have a single resultant.

6. For a system of forces all parallel to the same plane, show, by any method, that the sum of the moments is equal to the moment of the resultant round every line perpendicular to the plane.

7. For a system of forces all perpendicular to the same plane, show, by any method, that the sum of the moments is equal to the moment of the resultant round every line parallel to the plane.

8. Define the "centre of gravity" of a system of bodies; and investigate the values of its co-ordinates in terms of the co-ordinates and masses of the several bodies.

MR. LESLIE.

9. Prove that the centre of gravity of a trapezium divides the line joining the points of bisection of its parallel sides a, b in the ratio

26 + a 2a + b.

10. A trap-door turning on a hinge is supported by a weight attached to a string passing over a pulley in the same horizontal line as the hinge; find the reaction of the hinge.

11. A flexible string, acted on by gravity, only hangs in the form of a semicircle, its diameter being horizontal; find the law of variation of its density.

12. A flexible string passes over a rough curve; show how to find the tension at any point; and determine the limiting ratio of two weights connected by a cord passing over a rough vertical circle which is consistent with equilibrium.

C.

DR. STUBBS.

1. Show that the osculating circle of a curve has a contact of the third order at those points at which the radius of curvature is a maximum or a minimum.

2. Given two conjugate diameters in position; find the envelope of the ellipses whose area is constant.

3. In the curve ya + 2ay2x — ax3 = o, find the value of

gin of co-ordinates.

dy

at the ori

dx

4. Find the ordinate of the point in the curve y = a*, where the radius of curvature is a minimum.

MR. TOWNSEND.

5. The three distances of three points A, B, C in a right line, from a fourth point D in the line, being the three roots of the cubic

ax3 + bx2 + cx + d = 0,

form the reciprocal equation whose roots are the three reciprocal pairs of anharmonic ratios of the four points A, B, C, D.

6. Investigate, by any method, the number and signs of the real roots of the equation x1 - 6x3 + 9x2 + 8x-21=0.

7. If a and b be the spherical radii of any two circles of a sphere, c the spherical distance between their centres, and r the radius of the

sphere, required the area of the portion of the spherical surface common to both circles.

8. If a spherical triangle receive an infinitely small movement of rotation round the centre of its polar circle, show that the two derived triangles, determined by the connectors of corresponding vertices and by the intersections of corresponding sides of the original and displaced triangles, are in homology both with each of the latter and with each other.

MR. LESLIE.

9. A conic being represented by ax2 + 2bxy + by2 + 2gx + 2fy +c=0, determine (a) the equation of the tangents drawn through the origin; (b) the equation of the asymptotes.

10. Find the condition that

√ła + √mß + √ny = 0

should represent-—(a) a parabola, (b) a rectangular hyperbola.

11. If λα + β + νγ : = 0 and X'a + μ'ß + v'y = o, denote tangents to aa2+bB2 + cy2 + 2ƒßy + 2gay + 2haß = 0

and

a

h g

λ

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prove that the co-ordinates of the points of contact are

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the differentials being supposed to be taken with respect to one row or column only; and that the equation of the cord of contact is

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12. Form the equation of the reciprocal of S+ (λa +μB+vy)2, S being the most general equation of a conic; and hence show that if two conics have double contact, their reciprocals will also have double contact; and determine the equation of their common chord.

Logics.

STEWART'S ELEMENTS AND DISSERTATION.

MR. POOLE.

1. Show that the difficulties in the investigation of Metaphysics and Morals do not arise from the same sources as in Mathematics and Physics; and state some of the principal ones.

2. How does Stewart argue from the case of the equilibrist in support of his view as regards the question whether the mind can attend to several objects at once? Does his argument appear conclusive?

3. Explain the statement that "universals do not exist before thingsor after things-but in things."

4. Show that the principle, that we cannot by an effort of our will call up any given thought, is not contradicted by the power of recollection; and state what influence the mind possesses over the train of its thoughts.

5. How does Stewart argue on the question, whether the suspension of our voluntary operations arises from the suspension of the power of volition, or from the will losing its waking influence over us?

6. What apparent paradox does Stewart remark, arising from his theory, in the case of the memory of past events involving sensible objects; and what is his explanation?

7. What circumstances tend to increase the power of the memory?

8. Stewart states that Kant had been anticipated in his system by Cudworth. Show this; and state Cudworth's arguments in support of his views.

9. What are Kant's arguments to prove-first, the possibility, and next, the reality, of human liberty?

10. Stewart contrasts the conceptions of Des Cartes and of Leibnitz on the subject of "innate ideas." State them, and point out the superiority of Des Cartes ?

II. What is Collins' opinion as to the tendency of the doctrine of the liberty of man; and how does he argue in support of it?

12. What are Stewart's conceptions in reference to the idea of space? Compare them with those of Kant.

LOCKE AND COUSIN.

MR. ABBOTT.

1. Cousin states that Locke makes rewards and punishments the touchstone of moral rectitude; how does he establish this charge?

2. How does he show that rewards and punishments cannot be the foundation of moral order?

3. Locke denies that our senses show us only material things; on what grounds?

4. Cousin charges Locke with neglecting the evidence of induction ; discuss the justice of this charge.

5. "We are quite out of the way when we think that things contain in themselves the qualities that appear to us in them." Explain this remark, and the purpose for which Locke makes it.

6. Why does Locke "suspect that natural philosophy is not capable of being made a science"?

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