# State distinctly the conditions which must be satis- fied by the values of x which make u a maximum

## DIFFERENTIAL CALCULUS AND ANALYTICAL GEOMETRY

### a given sphere.

ORDINARY EXAMINATIONS, O.T. 1867. xlvii 6. Shew how to find the angle between the two

straight lines whose-equations are

### Ax + B y + C = 0 and A'x + B ' y + C = 0 .

7. The base and vertical angle of a triangle are given

find the equation to the locus of the vertex.

8. Shew that there is one sot of axes for which the term containing xy will disappear from the general equation, of the second degree: transform the equation xy = a2 to these axes.

9. Find the general equation to the tangent to the curve f { x , y ) = 0

Find the equation to the tangent to the curve xy = a2 at the point x = y = a

10. Any number of chords of an ellipse are drawn passing through a fixed point; shew that the intersections of tangents at their extremities will lie on a fixed straight line.'

11. Find the locus of the intersections of tangents to the hyperbola with perpendiculars drawn to them from the foci.

12. Shew that of any pair of conjugate diameters one only can meet the hyperbola: how are the ex- tremities of the other determined ?

13. Integrate the following functions :

2

### /

* 3 , /sin2x . dx

x V'Jax — x2 . dx I „ .,„

•' cos'a;

xlviii EXAMINATION PAPERS,

14. From the ends of the diameter of a circle as foci an ellipse is described whose minor axis, is half the diameter of the circle: find the area of the por- tion of the circle which is outside the ellipse.

15. A portion of a parabola bounded by the latus rectum revolves round the latus rectum : find the volume of the solid generated.

MIXED MATHEMATICS.—I.

{Professor Wilson.) 1. Find the conditions that the straight line

x — a _ y — / 3 _ z — y I m n

may lie wholly in tho plane A x + B y + Cz = D : 2. Define the hyperbolic paraboloid, find its equation

and shew that there are two systems of straight lines which wholly coincide with it.

3. Find the equation to the surface generated by the revolution of a circle whose radius is a round a line in its own plane at a distance {ii-\-b) from its center.

.4. Find the equation to the curve in which the surface in the preceding question is' cut by a tangent plane at a distance b from the axis of revolution.

ORDINARY EXAMINATIONS, O.T. 1867. x l i x

5. Define the osculating plane of a curve and find its equation.

6. A plane remaining constantly parallel to z = mx moves with uniform velocity v and from every point of its intersection with the plane of xy spheres continually originate whose radii at a tiine t after their origination are v't. Find the.

envelope of these spheres.

7. Shew that the moment of inertia of a plane area round any line perpendicular to its plane is the sum of the moments of inertia round two lines at right angles to one another in its plane through the point where the first line meets the plane.

8. From a sphere a portion is cut out by a cone whose axis passes through the center of the sphere and whose vertex is on the surface of the sphere : find the center of gravity of the remainder and its moment of inertia about the axis of the conical surface.

9. When any number of bodies move under their mutual attractions or repulsions shew that there is a plane determinable from the circumstances of the motion at any instant whose direction remains constant throughout the motion.

JO. Shew that when a solid is generated by,the revo- lution of a plane area round any axis in' its own plane which does not cut it the volume of the solid is equal to that of a cylinder whose base is the area and height the length of the path of its center of gravity.

1 . EXAMINATION P A P E R S ,

MIXED, MATHEMATICS.—II. • {Professor Wilson.)

1. Find the magnitude and line of action of the resultant of two parallel forces acting on a rigid body in opposite directions.

2. A flexible string in contact with a rough plane curve is on the point of motion : investigate the relation between the tensions of its free extremities. . * 3. Investigate the equation to the common catenary.

4. Every particle of a homogeneous spherical shell attracts with a force varying inversely as the square of the distance: calculate the attraction of the whole shell on an .external particle.

5. A sphere whose center of gravity does not coin- cide with its center of figure rests on an inclined ,plane of given roughness: the inclination of the plane is slowly increased:. find the conditions that rolling and sliding may begin together.

,6. Find the equation to the path of a projectile and shew that the velocity at any point is that which would be acquired in falling from the directrix.

7. A body moves under the action of a force tending towards a fixed point, investigate the polar equa- tions of motion.

ORDINARY E X A M I N A T I O N S , O.T. 1867. li

8. A sphere is placed in a rough hemispherical bowl of twice its radius: find the time of a small oscillation in one plane.

9. Shew that when two bodies impinge on- one another there is a loss of vis viva- unless the elasticity is perfect.

10. A sphere of cork is fixed at the end of a thin rod of iron whose direction passes through the center of the sphere: the iron being heavy enough to sink the cork the whole is supported under water by a string attached to the iron rod and to a float:

Find the condition that the rod-may be horizontal and determine whether the equilibrium is stable or otherwise.

11. Investigate a formula for finding the difference of height of two stations by means of the barometer, taking account of the variation of gravity.

12. Investigate the equation of continuity.

NATURAL PHILOSOPHY.

{Professor Wilson.)

Eight questions roust be answered correctly to entitle a Candidate to pass.

1. A pencil of rays diverging from a point midway between the center and the principal focus falls upon a concave spherical reflecting surface : inves- tigate the position of the focus of the reflected pencil.

c 2

lii EXAMINATION PAPERS,

2. Two plane mirrors are inclined to one another at an angle of 45°: draw a' figure shewing the course of a pencil by which a person standing between them sees his own profile after two reflexions, stating as in a proposition of Euclid how every line is drawn.

3. To a person looking vertically downwards into still water the apparent depth is four feet six inches : investigate the proper formula and calcu- late the true depth.

4. Investigate the general formula connecting the positions of the foci of incident and refracted rays in the case of a lens, assuming the requisite formula in the case of a single spherical refracting surface.

5. Describe Newton's fundamental experiment by which he shewed that common sunlight consists of a mixture of lights of different colours having different refrangibilities explaining the details by means of a carefully drawn diagram.

6. Describe and explain an experiment by which it is shewn that the colours of ordinary transparent coloured media are due to a stoppage only of a portion of the light and not to a modification of that which is transmitted.

7. Explain what is meant by fluorescence. and de- scribe and explain an experiment by which it is . shewn that when light is incident on a fluorescent surface a real modification of the light takes place.

8. Describe the construction of the ordinary stereo-

ORDINARY E X A M I N A T I O N S , O.T. 1867. l i i i

scope and explain by means of diagrams how the combination of two flat pictures produces the effect of objects in relief.

9. Describe an experiment by which it is shewn that two portions of light proceeding from the same point and arriving at the same point by different

paths may produce darkness. v

' 10. A small compass needle balanced on its middle point is placed in the same horizontal plane with a fixed horizontal long bar magnet at some dis- tance on one side of it and nearer to one end than the other : draw and explain a diagram showing the forces which determine the direction of the needle supposing the power of the bar magnet so great that the magnetic action of the earth may be neglected.

11. Explain how the horizontal components of the earth's magnetic force at two stations may be compared, supposing the magnetic state of the instruments used to have remained unaltered between the observations; and illustrate your explanation by diagrams.

12. Describe Wheatstone's experiment for ascertaining the velocity of propagation of electricity and the phenomena observed by him.

13. The current from a battery passes through a tan- gent galvanometer and produces a deflection of 60°; an additional resistance of a hundred feet of wire is then inserted in the circuit and the deflection is reduced to 3 0 ° : deduce from Ohm's law the united resistance of the battery and gal- vanometer in terms of a length of the wire used.

]1V EXAMINATION P A P E R S ,

14. The secondary terminals of a powerful induction coil are connected with the coil of a galvanometer and the coil put in action : what will be the effect on the needle of the galvanometer 1st when the circuit is metallic throughout 2nd when a small gap of air intervenes- and what inference is drawn from the difference ?

15. A current passes through a wire which is bent into a circle whose plane is vertical and which can move freely about a vertical diameter: wdiat posi- tion will the plane of the circle assume under the magnetic action of the earth, and in what direc- tion will the current traverse the circle ?

16. The Readings of the circle of a Transit Circle are as follows:

Horizontal wire coincident with

its reflexion ... ... 232° 10' Star at upper transit ... ... 16 12 The same star at lower transit ... 343 52 Find the latitude of the place and the polar dis- tance of the star.

17. The error of collimation of a transit instrument is 12": find the corresponding error in the time of transit of a star whose polar distance is 30°.

18. Explain the cause of the Astronomical correction known as the aberration of light and state ap- proximately its amount.

19. The Parallax of the Sun as deduced from Transits of Venus in 1761 and 1769 was assumed to be 8"-58; and as recently deduced from observations

O R D I N A R Y EXAMINATIONS, O.T. 1867. Iv

### 20. State Kepler's Laws and the inferences deduced

Outline

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