24. Case V. Let the primitive and the new axes have the same origin, but different directions. 25. In applying the preceding formulæ for the transformation of co-ordinates to particular cases, it will be necessary to affect the co-ordinates with the signs of direction which belong to their positions, and also to give to the functions of the angles, w', w", the signs which belong to them, in consequence of particular values of the angles, which, in order that the new may be capable of having every possible inclination to the primitive axes, may be any whatever between 0 and 360°. As the origin of the new axes may be situated in any of the angles round the origin of the primitive axes, it will be also necessary to consider x', y' as capable of having positive or negative signs, according to their position with respect to the origin of the primitive axes. As a particular case, illustrative of these observations, Let Oв==3, 0,B=y' =2, XOY=w=87° the reverse angle X"0,x,='240° Substituting these values of x', y', w, w', w", in the general formulæ, and observing that from the position of O, B, the sign of y' is negative, x = 3+2, sin. (87°—-240°)+y, sin. (87°—315°) x= y=-2+ sin. 87° x, sin. 240°+y, sin. 315° But (Trigon. Art. 19.), sin. 87° = = -sin. 27°; sin. (870-240°)=sin. (—153°)= − sin. (180°— 153°) = -sin. (180° - 228°)= + sin. 48°; = - sin. 60°; -sin. 45°. Making these substitutions in the values of x, y, 26. Transformation of rectilinear into polar co-ordinates. A line being traced on a plane, it may be proposed to determine a polar equation of that line, by assuming, in a convenient manner, the pole and the initial line. Commonly, however, it is supposed that the equation of the line is already determined by means of an equation between the rectilinear co-ordinates of its points, and it is required from this equation to deduce another expressed in terms of polar co-ordinates. To investigate general formula for the transformation of rectilinear into polar co-ordinates. Let oc or x, and PC or y, be the co-ordinates of any point P, referred to the rectilinear axes Ox, OY; also, let the angle X,O,P or 4, and the radius vector O, P, or r, be the polar co-ordinates of the same point. Through o, let o,x", o,y", be drawn parallel to ox, OY, respectively. Then XOY=w, the angle of the rectilinear axes; OB=x', O,B=y', the co-ordinates of the pole o,; and X"o,x,' the angle formed by the axis of x, and the initial line, are given: and it is required to express x, y, functions of w', x', y', w, r, and 0. in Because ox, 0,x" are parallel lines, Y P X X B X Y# and OY, BY", CP, are also parallel lines, X"0,Y"=X0Y=w, 0,PD=PO,Y"=X"0,Y"-X"0,x,-X,O,P=w—'—, 0,DP=180°-x" o,Y"=180°-w.. sin. O,DP sin. w. PO,D=X′′ 0,X,+x,0,P=w'+0. 1st. To find the expression for x. x=0C=OB+BC: are the most general formulæ for the transformation of rectilinear into polar co-ordinates. 27. Particular cases. Case I. Let the pole coincide with the origin of the rectilinear axes. In this case '=0; y'=0, and the formulæ become Case II. Let the initial line be parallel to the axis of x. Since the initial line is parallel to the axis of x, the angle w' =0, Case III. Let the pole coincide with the origin of the rectilinear axes and the initial line with the axis of r. In this case, x'=0; y'=0; w'=0 sin. w=1, sin. (--)=sin. (90°—(w'+0)=cos. (w' +0), The general formulæ become Case IV. x=x'+r cos. (w' + 0) y=y'+r sin. (w' +0). And the formulæ of Cases I. II. III. become respectively, SECTION II. NATURE AND PROPERTIES OF THE CONIC SECTIONS, OR CURVES OF THE SECOND ORDER. CHAPTER I. THE CONJUGATE DIAMETERS OF A CURVE OF 28. UNDER the name of conic sections are comprehended the curve lines which arise from the intersections of a plane with the curve surface of an upright cone; the point, the straight line, the isosceles triangle, and the circle, which result respectively from the plane touching the cone at the vertex, or on the curve surface, or from cutting it through the vertex, or in a direction parallel to the base (Geom. Cor. Def. 7. Cylinder, &c.) not being, in general, included. But in the present section a general equation with two variable quantities is taken to denote a relation between the co-ordinates of any point in a curve line of the second order; and from such equation, which is of the second degree, the forms and principal properties of the curves will be analytically investigated. It will be shown in the third section that the curves are identical with those which result from the actual sections of a cone by a plane surface. 29. The general equation has usually the form A'y'2 + B'x'2 + C' x'y' + D'y' + E'x'=F'. (A) in which each of the coefficients, A', B', &c., represents some quantity differing in value in the different positions which the line or rectangle represented by it may have; but in any particular positions such coefficients are constant, while the co-ordinates x and y of any point in the curve, with reference to them, are variable. The co-ordinate axes may be supposed to form any angle, right or oblique, with one another. 30. This equation may, in the following manner, be transformed into one in which the origin of the co-ordinates shall have any position, and in which the co-ordinate axes shall make any angles with the original axes. |