will be satisfied for every point of the line OP, and P3 P2 N' Y therefore, this equation is said to be the equation of the line OP. Conversely, if we were asked what locus was represented by the equation y = mx, У write the equation in the form =m, and the question is, "to find the locus of a point P, such that, if we draw PM, PN parallel to two fixed lines, the ratio PM: PN may be constant." Now this locus evidently is a right line OP, passing through O, the point of intersection of the two fixed lines, and dividing the angle between them in such a manner that sin POM= m sin PON. If the axes be rectangular, sin PON= cos POM; therefore, m=tan POM, and the equation y=ma represents a right line passing through the origin, and making an angle with the axis of x, whose tangent is m. 19. An equation of the form y=+mx will denote a line OP, situated in the angles YOX, Y'OX'. For it appears, from the equation y=+mx, that whenever x is positive y will be positive, and whenever x is negative y will be negative. Points, therefore, represented by this equation, must have their co-ordinates either both positive or both negative, and such points we saw (Art. 3) lie only in the angles YOX, Y'OX'. On the contrary, in order to satisfy the equation y=-mx, if x be positive y must be negative, and if x be negative y must be positive. Points, therefore, satisfying this equation, will have their co-ordinates of different signs; and the line represented by the equation, must, therefore (Art. 3), lie in the angles Y'OX, YOX'. 20. Let us now examine how to represent a right line PQ, situated in any manner with regard to the axes. Draw OR through the origin parallel to PQ, and let the ordinate PM meet OR in R. Now it is plain (as in Art. 18), that the ratio RM: OM will be always constant (RM always equal, sup Y P pose, to m.OM); but the ordinate PM differs from RM by the constant length PR=0Q, which we shall call b. Hence we may write down the equation that is, PM=RM+ PR, or PM=m.OM+PR, y=mx+b. The equation, therefore, y = mx + b, being satisfied by every point of the line PQ, is said to be the equation of that line. It appears from the last Article, that m will be positive or negative according as OR, parallel to the right line PQ, lies in the angle YOX, or Y'OX. And, again, b will be positive or negative according as the point Q, in which the line meets OY, lies above or below the origin. Conversely, the equation y = mx + b will always denote a right line; for the equation can be put into the form Now, since if we draw the line QT parallel to OM, TM will beb, and PT therefore y-b, the question becomes: "To find the locus of a point, such that, if we draw PT parallel to OY to meet the fixed line QT, PT may be to QT in a constant ratio;" and this locus evidently is the right line PQ passing through Q. The most general equation of the first degree, Ax+By+C=0, can obviously be reduced to the form y=mx+b, since it is equivalent to A C y X B Bi this equation therefore always represents a right line. 21. From the last Articles we are able to ascertain the geometrical meaning of the constants in the equation of a right line. If the right line represented by the equation y=mx+b make an angle =a with the axis of with the axis of y, then (Art. 18) and and if the axes be rectangular, m = tana. We saw (Art 20) that b is the intercept which the line cuts off on the axis of y. If the equation be given in the general form Ax+By+C=0, we can reduce it, as in the last Article, to the form y=mx+b, and we find that or if the axes be rectangular tana; and that = B length of the intercept made by the line on the axis of y. is the COR. The lines y=mx+b, y=mx+b' will be parallel to each other if m=m', since then they will both make the same angle with the axis. Similarly the lines Ax + By + C = 0, A'x + B'y + C' = 0, will be parallel if Beside the forms Ax + By +C=0 and y=mx+b, there are two other forms in which the equation of a right line is frequently used; these we next proceed to lay before the reader. D 22. To express the equation of a line MN in terms of the intercepts OM-a, ON=b which it cuts off on the axes. We can derive this from the form already considered A B Ax+By+C=0, or C x+y+1=0. This equation must be satisfied by the co-ordinates of every point on MN, and there fore by those of M, which (see Art. 2) are x=α, y=0. Hence we have Substituting which values in the general form, it becomes This equation holds whether the axes be oblique or rectangular. It is plain that the position of the line will vary with the signs of the quantities a and b. For example, the equation х +2=1, which cuts off positive intercepts on both axes, re a b presents the line MN on the preceding figure; -=1, cutting α off a positive intercept on the axis of x, and a negative intercept on the axis of y, represents MN'. By dividing by the constant term, any equation of the first degree can evidently be reduced to some one of these four forms. Ex. 1. Examine the position of the following lines, and find the intercepts they make on the axes: 2x-3y=7; 3x+4y+ 9 = 0; 3x+2y= 6; 4y-5x=20. Ex. 2. The sides of a triangle being taken for axes, form the equation of the line joining the points which cut off the mth part of each, and show, by Art 21, that it is parallel to the base. Ans. x У 1 23. To express the equation of a right line in terms of the length of the perpendicular on it from the origin, and of the angles which this perpendicular makes with the axes. Let the length of the perpendicular OP=p, the angle POM which it makes with the axis of x = a, But ? = cosa, a N P M = cos; therefore the equation of the line is x cosa+y cosẞ=p. In rectangular co-ordinates, which we shall generally use, we have ẞ=90°-a; and the equation becomes a cosa + y sin a=p. This equation will include the four cases of Art. 22, if we suppose that a may take any value from 0 to 360°. Thus, for the position NM', a is between 90° and 180°, and the coefficient of x is negative. For the position M'N', a is between 180° and 270°, and has both sine and cosine negative. For MN', a is between 270° and 360°, and has a negative sine and positive cosine. In the last two cases, however, it is more convenient to write the formula x cosa+y sina: -p, and consider a to denote the anglé, ranging between 0 and 180°, made with the positive direction of the axis of x, by the perpendicular produced. In using then the formula x cosa + y sin a=p, we suppose p to be capable of a double sign, and a to denote the |