Such symbols as 4, O, II, &c., are written <s, Os, Ils, when used in the plural.

Of course if the student can write out any word as quickly and as distinctly as he can write its symbol there is no absolute need for him to use the symbol; but if he once begin to use the symbol he should use it consistently throughout, lest he should both confuse himself and the examiner who has to read after him.

ANALYTICAL OR ALGEBRAICAL GEOMETRY.

"Algebra is applied to Geometry to investigate problems which concern the magnitudes of lines or areas, or to express the position of points and the form of curves."-G. H. Puckle, M.A.

"In Analytical Geometry we connect the sciences of space and number; we determine equations which represent certain well-known lines and curves, and deduce their geometrical properties from those equations; or, having given equations, we discuss the nature of the curves represented by them." T. G. Vyvyan, M.A.

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Analytical Geometry, or Algebraical Geometry, as it is sometimes called, is the application of Algebra to the investigation of the properties of geometrical lines and surfaces. It shows that all such lines and surfaces may be as significantly represented by algebraical equations as by geometrical figures, and that the various properties of them may be deduced from these equations by the ordinary processes of common Algebra. As at the outset of the investigation the geometrical conditions are translated into Algebra, so the successive algebraical deductions may be translated back into geometrical theorems-the necessary consequences those conditions. The subject is as interesting as it is useful; and there is, perhaps, no department of mathematics which, to the zealous student, possesses such attractiveness as Analytical Geometry."-J. Hann.

"The remarkable contrivance, first introduced by Descartes, of representing lines and surfaces by algebraical equations was admirably calculated to extend our knowledge of the properties of figured space; for, as we are thus enabled to embody in such an equation every property and peculiarity belonging to any curve or surface as soon as we know its distinguishing characteristics, or the law of its description, it is plain that, in order to develope these several properties in succession, nothing more is necessary than a careful analysis of the algebraical expression in which they are all implicitly contained."-J. R. Young.

ELEMENTS OF CO-ORDINATE GEOMETRY, AS FAR AS THE EQUATIONS AND PROPERTIES OF THE RIGHT LINE AND CIRCLE. This part of the Regulations represents a change which implies that, although the number of subjects specified has been reduced, more thoroughness and wider application are required in what is now stated in more general terms. The candidate should not be slow to appreciate the full meaning of the change. He has not now to study the equations to what are strictly called the Conic Sections; but the restriction which was formerly attached to the requirement with regard to the Straight Line and Circle is now removed. Formerly their equations were to be "referred to rectangular co-ordinates" only; at present the whole subject of Plane Co-ordinate or Analytical Geometry is required as far as it applies to the Straight Line and the Circle, which must now have their equations and properties investigated more extensively in reference to both Oblique Co-ordinates* and Polar Co-ordinates, as well as to those which are Rectangular.

*This does not now apply to the stricter definition of this subject which the University has just recently and very wisely given. "Rectangular and Polar Co-ordinates" only are now required for the Pass Examination. The Honours Examination remains the same.

Upon this subject it is of first importance that the student should acquire lucid and definite ideas. He should ascertain the meaning of every step he takes in all its various capabilities of investigation. He must learn to reason as a geometrician whilst employing the skill and analytical power of the practical algebraist. It is, however, a beautiful and fascinating study, and the student will soon find himself en rapport with it, especially if he begins by a clear understanding of its elementary definitions, and of its main principles of operation in forming and analysing equations. It will be impossible for us within the limits of this work to present an exhaustive outline of the subject. Nor is it necessary. The text-books which

we shall presently describe contain an ample and intelligible treatment of every part required. We may, however, call attention to some of the principles, processes, and main propositions with which the candidate should be thoroughly familiar.

ELEMENTARY PRINCIPLES.

1. The principle upon which the use of Polar Co-ordinates proceeds is that, The position of a point in a plane is known when we can measure its distance from a fixed point, and the inclination of this distance to a fixed line passing through the given point. And since a straight line or circle, represented by an equation between the polar co-ordinates, is the assemblage of all those points whose co-ordinates satisfy the equation, we may determine the straight line or circle by means of those co-ordinates.

2. The principle upon which the ordinary, or Cartesian, system of co-ordinates is based is that, The position of a point is known when we can measure its distances from two fixed straight lines which intersect. In the same way as before: Since a straight line or circle, represented by an equation between the co-ordinates of this system, is the assemblage of all those points whose co-ordinates satisfy the equation, we may trace the locus of the straight line or circle by means of the points represented by those co-ordinates.

3. The fundamental idea of the whole subject is the relation between a locus and an equation. The following different expressions of the same essential principle may make the idea clearer to the minds of our readers and help them to grasp it in the thorough way that is absolutely necessary.

The equation to a line straight or curved is that equation which expresses the invariable relation which exists between the co-ordinates of every point of the line; for a line may be regarded as an assemblage of points which make up the line. That is, the points may be brought so indefinitely near to each other as to form a continuous line. The "invariable relation" is one which is satisfied by the

co-ordinates of every point of the line. Hence the equation may be considered as representing the line which passes through all the points whose co-ordinates satisfy it.

This line, straight or curved, is called the locus of the said equation, because the co-ordinates of every point on the line satisfy the equation. For example, if the equation y-x-2=0 be given, suppose x=1, then y=3. These values denote the position of a point in a certain line represented by the equation. We might strictly call this position of the point the locus of the point. But the word has a wider and a more conventional meaning. Suppose, for instance, we give to x in succession the values 1, 2, 3, 4. we get for corresponding values of y, 3, 4, 5, 6 . . . . Each pair of values, such as 1 and 3, 2 and 4, 3 and 5, &c., denotes a point in the line represented by the equation. If we imagine, as we clearly may, the points denoted by these values to be so near to each other as to form a continuous line, this line is called the locus of the equation. In other words: The line which passes through every point determined by giving to x and y values that satisfy the equation y-x2=0, is called the locus of the equation.

=

The student may see at a glance that the above equation represents a straight line. But the above reasoning is perfectly general, and will apply equally well if we use instead the equations x3+ y2=c2, x2 y2 y2 = 4ax, = 1, which are the equations to the circle, parabola, ellipse, and hyperbola respectively. But in these cases, for every possible value of x we get two values of y equal in magnitude but opposite in sign, showing the symmetrical structure of the curve with respect to the axis of x.

The following is another illustration of the meaning of the term locus: "The locus of a point varying in position is the line, whether straight or curve, in which the point, in all its positions, is placed; so that when we enquire what is the locus of a point satisfying certain conditions we ask for the determination of the line which the varying point traces out. The present problem might be otherwise proposed in the following terms, namely: From a point between two given straight lines a perpendicular to each line is drawn; these perpendiculars are equal. What is the locus of the point? The answer would be that the locus is a straight line passing through the point of junction of the two given lines if they meet, and parallel to them if they themselves are parallels. It is an obvious inference from the fact that the locus is a straight line and that each point in it is equidistant from the given lines, that it bisects the angle between the sides of which every point in the line is posited."

Outline of the Principal Propositions and Formulae.

THE POINT.

1. To find an analytical expression for the distance, D, between two points (x1 y1), (X2 Y2).

For rectangular co-ordinates the expression is

D= {(x2-x1)2+(y 2 − Y 1)2 } *.

For oblique co-ordinates

D = {(x2 − x1)2 + (Y 2 − Y 1)2 + 2(x, − x1) (Y2 - Y1) cosw}}. For polar co-ordinates D={r2+r22-2r1r2 cos(01~02 ) } }

2. To find the co-ordinates of a point (hk) which divides in a given ratio (m:n) the straight line joining two points ̧(x1Y1), (X2 Y2).

When the straight line is to be bisected, then m=n, and we have ; = X 2 + x1, k = Y 2+ Y1.

h=2
2

2

3. To find an analytical expression for the area of a triangle in terms of the co-ordinates of its angular points.

Area =x21-X1Y2+X ̧Y2−X2Ys+X1Ys−XsY1}·

To remember this formula, observe that x is in the order of 2, 1, 3; that y goes with it in symmetrical pairs throughout; and that the signs are alternately positive and negative. We may also note that y with the positive sign has the natural order, 1, 2, 3.

If the axes be oblique and inclined at an angle w, it is easy to see that the preceding expression must be multiplied by sin. w.

With polar co-ordinates there are two expressions in which the co-ordinates must be taken in such an order as to make the whole expression positive.

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Area r2r, sin(02 – 03)—r1⁄2rı sin(0, -01)—rır, sin(01 −0 ̧)¦ =r2r, sin(02-03)+r1r sin(01-02)+rr1 sin(0, -01)}.

TRANSFORMATION OF CO-ORDINATES.

By this is meant an investigation of the definite relations connecting the different systems of co-ordinates. When we know the co-ordinates of a point with regard to any origin and axes, we can obtain the co-ordinates of the same point with regard to any other origin and axes. An equation may often be put in a more simple form by changing the origin or the axes, or both. For example: the equations to a straight line or a circle assume a simpler form