the new axis of X makes with the old, XOx = a; let YOx = ẞ: then the formulæ become 10. Lastly, by combining the transformations of the two preceding articles, we can find the co-ordinates of a point referred to two new axes in any position whatever. We first find the coordinates (by Art. 8) referred to a pair of axes through the new origin parallel to the old axes, and then (by Art. 9) we can find the co-ordinates referred to the required axes. The general expressions are obviously obtained by adding x and y' to the values for x and y given in the last article. Ex. 1. The co-ordinates of a point satisfy the relation x2 + y2 – 4x what will this become if the origin be transformed to the point (2, 3)? 6y = 18; Ans. X2+ Y2 = 31. Ex. 2. The co-ordinates of a point to one set of rectangular axes satisfy the relation y2 - x2 6: what will this become if transformed to axes bisecting the angles between the given axes? Ans. XY = 3. = = Ex. 3. Transform the equation 2x2 - 5xy + 2y2 4 from axes inclined to each other at an angle of 60°, to the right lines which bisect the angles between the given axes. Ans. X2 27Y2 + 12 = 0. Ex. 4. Transform the same equation to rectangular axes, retaining the old axis of x. Ans. 3X210Y2 – 7XY√3 = 6. Ex. 5. It is evident that when we change from one set of rectangular axes to another, x2 + y2 must X2+ Y2, since both express the square of the distance of a point from the origin. Verify this by squaring and adding the expressions for X and Y in Art. 9. Ex. 6. Verify in like manner in general that 11. The degree of any equation between the co-ordinates is not altered by transformation of co-ordinates. Transformation cannot increase the degree of the equation: for if the highest terms in the given equation be am, ym, &c., those in the transformed equation will be {x'sino + x sin(w−a) + y sin(w−ẞ)}", (y' sin w + x sina + y sin ß)m, &c., which evidently cannot contain powers of x or y above the mth degree. Neither can transformation diminish the degree of an equation, since by transforming the transformed equation back again to the old axes, we must fall back on the original equation, and if the first transformation had diminished the degree of the equation, the second should increase it, contrary to what has been just proved. 12. Polar Co-ordinates.-Beside the method of expressing the position of a point which we have hitherto made use of, there is also another which is often em ployed. If we were given a fixed point O, and a fixed line through it, OB, it is evident that we should know the position of any point, P, if we 0 knew the length OP, and also the P B angle POB. The line OP is called the radius vector; the fixed point is called the pole; and this method is called the method of polar co-ordinates. It is very easy, being given the x and y co-ordinates of a point, to find its polar ones, or Y For the more ordinary case of rectangular co-ordinates, w = 90°, and we have only to substitute 0-a for in the preceding formulæ. C For rectangular co-ordinates we have x = p cos(-a) and y = p sin (0 - a). Ex. 1. Change to polar co-ordinates, the following equations in rectangular coordinates. Ans. p = 5m cos9. Ex. 2. Change to rectangular co-ordinates the following equations in polar co-ordi nates. x2 + y2 = 5mx. (x2 + y2)2 = a2(x2 — y2). Ans. x2 + y2 = (2a − x)2. 13. To express the distance between two points, in terms of their polar co-ordinates. 14. WE saw, in the last chapter, that we could determine the position of a point, being given two equations regarding its coordinates, of the form x = a, y = b. It is evident that we could equally determine the point, had we been given any two equations of the first degree between its co-ordinates, such as Ax+ By + C = 0, A'x + By + C' = 0, for we have here two equations between two unknown quantities, which we can solve by eliminating y and x alternately between them, and obtain two results of the form x = a, y = b. Ex. What point is denoted by the equations 3x+5y = 13, 4x - y = 2? Ans. x = 1, y = 2. 15. Two equations of higher order between the co-ordinates would represent, not one, but a determinate number of points. For, eliminating y between the equations, we obtain an equation containing x only; let its roots be a1, a2, as, &c. Now, if we substitute any of these values (a,) for x in the original equations, we get two equations in y, which must have a common root (since the result of elimination between the equations is rendered by the supposition x = a1). Let this common root be y = B1. Then the point whose co-ordinates are x = a1, y = B1, will at once satisfy both the given equations; and so, in like manner, will the point whose co-ordinates are x = a2, y = B2, &c. = 0 If the given equations were of the mth and nth degrees respectively, the equation in x would (by the theory of elimination, see Lacroix's Algebra, § 196, p. 278; Young's Algebra, § 124, p. 229) .be of the mnth degree, and consequently there would be mn roots a1, a2, &c., and, therefore, mn points represented by the two equations. x2 + y2 = 5, Ex. 1. What points are represented by the two equations Eliminating y between the equations, we get x4 5x2 + 4 = 0. equation are x2 = 1 and x2 = 4, and, therefore, the four values of x are of y, x = + 1, x = - - 1, x = + 2, x = - 2. Substituting any of these in the second equation, we obtain the corresponding values The two given equations, therefore, represent the four points Ex. 2. What points are denoted by the equations Ex. 3. What points are denoted by the equations x2 - 5x + y + 3 = 0, x2 + y2 — 5x-3y+ 6 = 0 ? Ans. (4, 8), (− 3, − 4). Ans. (1, 1), (2, 3), (3, 3), (4, 1). 16. Having seen that any two equations between the co-ordinates represent geometrically one or more points, we proceed to inquire the geometrical signification of a single equation between the co-ordinates. We shall find the case to be similar to the solution of a class of geometrical problems, with which the learner is familiar. We can determine a triangle, being given the base and any other two conditions, but had we been given only one other condition, the vertex, though no longer determined in position, would still be limited to a certain locus. So we shall find, that although one equation between the two co-ordinates is not sufficient to determine a point, it is, however, sufficient to limit it to a certain locus. In fact, the equation asserts, that a certain relation subsists between the co-ordinates of every point represented by it. Now, although this relation will not in general subsist between the co-ordinates of any point taken at random, yet there will be more points than one for which this relation will be true; the assemblage of these points will form a locus of points whose co-ordinates satisfy the equation, and this locus is considered the geometrical signification of the given equation. Y K P' P That a single equation between the co-ordinates signifies a locus, we shall first illustrate by the simplest example. Let us recall the construction by which (p. 2) we determined the position of a point from the two equations x = a, y = b. We took OM a; we drew MK parallel to OY; and then, measuring MP=b, we found P, the point required. Had we been = M X given a different value of y, x = a, y=b, we should proceed x as before, and we should find a point P' still situated on the line MK, but at a different distance from M. Lastly, if the value of y were left wholly indeterminate, and we were merely given the single equation = a, we should know that the point P was situated somewhere on the line MK, but its position in that line would not be determined. Hence the line MK is the locus of all the points represented by the equation x = a, since, whatever point we take on the line MK, the a of that point will always = a. 17. In general, if we were given an equation of any degree between the co-ordinates, let us assume for r any value we please (xa), and the equation will enable us to determine a finite number of values of y answering to this particular value of x, and, consequently, the equation will be satisfied for each of the points (p, q, r, &c.), whose x is the assumed value, and whose y is that found from the equation. Again, assume for x any other value |