results fzdy = TX2 for the whole solid above the plane xY, and consequently the whole fluent of zdy = TX2 = Ty2 if y is the ordinate of the solid of the revolution; and u, which =ffzdydx, = fry2dx. CHAPTER X. Spirals. 1. Def. 1. A curve which is generated by a variable line revolving in the same plane about a fixed point is called a spiral. The spiral is, in general, determined and its properties ascertained from the relation which: subsists between the revolving line, which is called the radius vector, and the angle described. Let sp the radius vector = r, (vid. fig. 4) ▲ ASP = 0, rad. = 1. With radius sa describe a circle AD cutting SP in D; then AP may be supposed to be generated by an ordinate as DP, which produced always passes through s, moving along the abscissa AD; or SP, which = SD + DP, may be considered as the function and AD its independent variable. Def. 2. s is the pole of the spiral, and the equation which expresses the relation between SP and the ASP is called its polar equation. SP and AD are the polar co-ordinates. Some writers characterize the spiral by the equation between sp and a perpendicular drawn from s upon a tangent at P. 2. Examples. Ex. 1. Let AP be the Apollonian parabola, a the vertex, its focus being the pole, SA = a, sy perpendicular on the tangent at P = = p, then r = 2a -r cos.0 and p2 = ar are its two equations, the first of which is its polar equation. Ex. 2. A right line as TV may be considered as a spiral. For take s a known point round which SP revolves, so that P describes the line TPV; let the fixed < T = α, PST= ; .. ST: SP:: sin.(a + ◊): a. sin.a sin.a; or r= sin.(a + 0) considered as a spiral. V P T a a is the polar equation to a right line 3. Given the equation of a curve between its rectangular co-ordinates; required its polar equation. Let the pole s be taken as the origin, and let sa the first radius of the spiral be the axis of the abscissæ; SN = x, NPY, SP = r, 4 NSP = ; then xr cos. 0, y = r sin.0; and these being substituted in the equation between x and y, give the required equation between r and 0. Cor. 1. Conversely, if the polar equation be given, the equation between the co-ordinates may be found by substituting for r and §, r = √x2+ y2, cos. ==, sin.8 = 2, y tan. 9 = &c. &c. as the case may require. r Cor. 2. If it be also required to change the direction of the axes, must be substituted for in the above expressions, being the angle at which the new axis x is inclined to SA. Cor. 3. If the angle enters into the polar equation, as in the example r = a.9", the equation of the rectangular coordinates cannot appear under an algebraick form; for the arc is not an algebraick function of its sine or cosine. In this case, by differentiation and elimination, de may be obtained in terms of x, y, dx and dy. 4. Examples. Ex. 1. In a parabola y = 4ax, or transferring the origin to the pole, y2 = 4a(a + x) .. by substitution 4a2 + 4ar cos.02 sin.20 = r2(1-cos.20) .. 4a2 + 4ar cos. +r2 cos.20 = r2:. 2a = r cos. = r; or r = 2a — r cos. is the required equation, cos. being negative when is greater than 90°. 1 b2 = 1, or COS.20 sin.20 + a2 is the re centre as the pole, and the semi-axis major as the first radius, r2 cos.20 r2 sin.20 quired equation. Cor. Hence it may be shown that when the eccentricity is small, the excess of the radius vector above the semi-axis minor cos.20 nearly. αν acos.20; substitute xr-b, or r=x+b, Ex. 3. Required the polar equation of an ellipse when the focus is the pole. b2 = (a2-x2). Also r2 = sp2 = SN2 + Np2 = (CN ~Cs)2 a2 + NP2 = (x~ae)2 + (1−e2) (a2 — x2) = a2 - 2αex + e2x2 ..rma- ex =, by substitution, a -e (ae + r. cos.) a(1 − e2) er cos., and consequently r = a.(1-e1) = where 1+e cos.0 cos. is negative when ASP is greater than 90°. Cor. If the 4 is reckoned from the greatest radius Ex. 4. Required the polar equation of a circle when the pole is in its circumference. y2 = 2ax-x2.. r2sin.0 = 2ar cos. r2 cos.20 =2a cos. is the required equation. 5. Required geometrical representations for the fluxions of the spiral, the radius vector and the angle described. R N With radii SP, SA describe the circular arcs PV, AD; draw PQ a tangent to the circle at P or perpendicular to sp, and take it to represent the fluxion of VP; draw QR at right angles to PQ meeting a tangent to the spiral at P in R; and draw saps near to SP; then, since spa is ultimately parallel to QR, it may be shown as in Ch. 8. Art. 2, that the sides of the curvilinear figure Pπp are ultimately in the ratio of the sides of the triangle PQR; whence PR and QR represent the fluxions of AP and SP on the scale that PQ = the fluxion of VP. Let SA = 1 AP S ASP = 0 Cor. If the then VP = SP.AD = r◊; and we have ASD be supposed to increase uniformly, PQ the base of the fluxional triangle must be made to vary as the radius vector. 6. Required to calculate the subtangent and the subnormal. Def. Through s draw T'sN' perpendicular to SP, meeting the tangent in T' and the normal in N'; then ST' is the subtangent, and SN' the subnormal. First, to calculate the subtangent. From which expressions the values of the subtangent and subnormal may be calculated, if the polar equation be given. 7. Required the perpendicular on the tangent in terms of the radius vector and the angle described. |