will be found by multiplying the expressions given above by sin w.

However the relative situations of A, B, C may be changed, the student will always find for the area of the triangle the expression (2), or that expression with the sign of every term changed. Hence we conclude, that we shall always obtain the area of the triangle by calculating the value of the expression (2), and changing the sign of the result if it should prove negative.

Locus of an equation. Equation to a curve.

...

y.

12. Suppose an equation to be given between two unknown quantities, for example, y-x-2=0. We see that this equation has an indefinite number of solutions, for we may assign to x any value we please, and from the equation determine the corresponding value of y. Thus corresponding to the values 1, 2, 3, ... of x, we have the values 3, 4, 5, of Now suppose a line, straight or curved, such that it passes through every point determined by giving to x and y values that satisfy the equation y-x-2=0; such a line is called the locus of the equation. It will be shewn in the next chapter that the locus of the equation in question is a straight line. We shall see as we proceed that generally every equation between the quantities x and y has a corresponding locus.

But instead of starting with an equation and investigating what locus it represents, we may give a geometrical definition of a curve and deduce from that definition an appropriate equation; this will likewise appear as we proceed; we shall take successively different curves, define them, deduce their equations, and then investigate the properties of these curves by means of their equations. We shall in the next chapter begin with the equation to a straight line.

The connexion between a locus and an equation is the fundamental idea of the subject and must therefore be carefully considered; we shall place here a formal definition which we shall illustrate in the next chapter by applying it to a straight line.

DEF. The equation which expresses the invariable relation which exists between the co-ordinates of every point of

curve, the

a curve is called the equation to the curve; and the co-ordinates of every point of which satisfy a given equation, is called the locus of that equation.

13. The student has probably already become familiar with the division of algebraical equations into equations of the first, second, third... degree. When we speak of an equation of the nth degree between two variables we mean that every term is of the form Axay where a and B are zero or positive integers such that a+B is equal to n for one or more of the terms but not greater than n for any term, and A is a constant numerical quantity; and the equation is formed by connecting a series of such terms by the signs + and −, and putting the result = 0.

EXAMPLES.

1. Find the polar co-ordinates of the points whose rectangular co-ordinates are

(1) x=1, y=1;

(3) x=-1, y=1;

(2) x=-1, y=2;

(4) x=-1,y=-1;

and indicate the points in a figure.

2. Find the rectangular co-ordinates of the points whose

polar co-ordinates are

4. Find the area of the triangle formed by joining the first three points in question 1.

5. A is a point on the axis of x and B a point on the axis of y; express the co-ordinates of the middle point of AB in terms of the abscissa of A and the ordinate of B; shew also that the distance of this point from the origin = } AB.

6. Transform equation (2) of Art. 11 so as to give an expression for the area of a triangle in terms of the polar co-ordinates of its angular points. Also obtain the result directly from the figure.

7. A and B are two points and O is the origin; express the area of the triangle AOB in terms of the co-ordinates of A and B, and also in terms of the polar co-ordinates of A and B.

8. A, B, C are three points the co-ordinates of which are expressed as in Art. 11; suppose D the middle point of AB; join CD and divide it in G so that CG=2GD; find the co-ordinates of G.

9. Shew that each of the triangles GAB, GBC, GAC, formed by joining the point G in the preceding question to the points A, B, C, is equal in area to one-third of the triangle ABC. See Art. 11.

the origin

10. A and B are two points; the polar co-ordinates of A are 0,, r and those of B ̄are 0, r2. A line is drawn from bisecting the angle AOB; if C be the point where this line meets AB shew that the polar co-ordinates 2rr, cos (0,0) of Care 0 = (0,+02) and r=

12

11. Find the value of CD and AD in question 8 in terms of the co-ordinates there used; and shew that

AC2+BC2=2 CD2 + 2AD2.

12. Find the value of GA, GB, and GC, in question 9 in terms of the co-ordinates there used; and shew that

3 (GA3 + GB3 + G.C2) = AB2 + BC2 + CA3.

CHAPTER II.

ON THE STRAIGHT LINE.

14. To find the equation to a straight line.

We shall first suppose the line not parallel to either axis. Let ABD be a straight line meeting the axis of y in B. Draw a line OE through the origin parallel to ABD. In ABD take any point P; draw PM parallel to OY, meeting OX in M and OE in Q.

Suppose OB=c, and the tangent of EOX=m; and let x, y be the co-ordinates of P; then

y=PM = PQ+ QM

= OB+ QM

=c+ OM tan QOM

= c + mx.

Hence the required equation is

y = mx + c.

OB is called the intercept on the axis of y; if the line

crosses the axis of y on the negative side of O, c will be negative.

We denote by m the tangent of the angle QOM or BAO, that is, the tangent of the angle which that part of the line which is above the axis of x makes with the axis of x produced in the positive direction. Hence if the line through the origin parallel to the given line falls between OY and OX, m is the tangent of an acute angle and is positive; if between OY and OX produced to the left, m is the tangent of an obtuse angle and is negative. So long as we consider the same straight line m and c remain unchangeable, they are therefore called constant quantities or constants. But x and y may have an indefinite number of values since we may ascribe to one of them, as x, any value we please, and find the corresponding value of y from the equation y=mx+c; x and y are therefore called variable quantities or variables.

If the line pass through the origin, c=0, and the equation becomes

y=
= mx.

15. We have now to consider the cases in which the line is parallel to one of the axes.

If the line be parallel to the axis of x, m = 0, and the equation becomes

y = c.

If the line be parallel to the axis of y, m becomes the tangent of a right angle and is infinite; the preceding investigation is then no longer applicable. We shall now give separate investigations of these two cases.

axes.

To investigate the equation to a line parallel to one of the