Ex. 2. A sphere of radius a is intersected by a right circular cylinder whose diameter is a and which passes through the centre of the sphere ; find the length of the curve of intersection. The equations of the sphere and the cylinder are respectively r = a, and p = a coso; so that sin 0 = cos ; and consequently s = 4a * {1+(cos p)2}} dø on comparing which with the expression for the perimeter of an ellipse given in (33), Art. 158, it appears that the length of the curve is equal to that of an ellipse of which the major axis = 2a, and the eccentricity = Ex. 3. Determine the length of the curve of intersection of the right cone, whose equation is 0 = a, with the skew helicoid whose equation is r cos 0 = ko. s = k tan a {(cosec a)2 + ?}dø k tanas = "1" } ${(cosec a)2 +$2}++(cosec a)? log ø+{\cosec alo + $43"}. cosec a This problem may also be stated in the following form. Find the length of the path described by a point which moving on the surface of a right circular cone uniformly recedes from the vertex while its meridian plane revolves uniformly. In this aspect the equations are, r = at, 0 = a, $ = bt, where t is a variable, and a and b are constants; and the expression for s is of the same form as the preceding Ex. 4. Find the equation to that curve on the surface of a sphere which cuts all the meridian lines at the same angle a. Let the radius of the sphere = a; then as add is the projection of ds on the meridian line, the differential equation of the curve is ado = cos a ds; whence we have do :: tan = escota, the limits being such that 0 = 5, when p=0. Hence from the differential equation of the curve, if s is measured from the point where 0 = ,0 = 0, s = a sec ale -); so that when the curve reaches the pole and 6 = 7,8 = sec a; · in which case $ = 0; whence it appears that the number of revolutions made by the curve on the surface of the sphere is infinite. The curve is called the Loxodrome. SECTION 4.—Investigation of various properties of Curves de pending on the length of the Arc and other quantities in terms of which the equation is expressed. The Intrinsic Equation of a Curve, 166.] In the preceding sections we have expressed the length of a curve between two given points, in terms of the coordinates of those points or at least of one of the coordinates. I propose now to investigate other theorems of curves depending on the length of the arc between two given points, and certain properties of the curve at these points; and in the first place I will take the inverse problem, and find the equation of a curve in terms of x and y, when an equation is given in terms of s and of one or both of the coordinates x and y. Let us suppose the relation between s, x, and y to be given in the explicit form s = f (x, y); (73) (74) (75) which will give an equation in terms of x, y, dx, dy; and from which in many cases the integral equation may be deduced by the processes already explained; in other cases processes will be required, which belong to a more advanced part of our treatise. Ex. 1. Let $ = x cos a +y sin a+c; . ds = dx cos a + dy sin a; dx2 + dy2 = (cos a)2 dx2 + 2 sin a cos a dx dy + (sin a)2 dy2. (sin a) dx? — 2 sin a cos a dx dy +(cos a)dy2 = 0, sin a dx - cos a dy = 0; PRICE, VOL. II. ff (77) | 2% 2% which is the equation to a straight line passing through the origin; and which evidently satisfies the given relation. (78) .. y = a versin-1 + (2 ax — 22); which is the equation to a cycloid, whose vertex is at the origin, and the radius of whose generating circle is a. The problem is evidently the inverse one to that of Ex. 3, Art. 155. Ex. 4. If 33 = ax?, the equation to the curve in terms of x and y is x +yš = ks, where 4 aš = 9k. This example is a particular case of a more general problem. Determine the values of m and n, when the equation to a curve can be expressed in finite terms of x and y, the defining property of the curve being sm+ h = am 2". Ex. 5. If ga = ya — a’, the curve is the catenary whose equation is y leáte . : = a cos 0; which is the equation to a circle, whose diameter is a, and whose pole is at a point on the circumference. Ex. 7. If 8 = a My, the equation to the curve is Ex. 9. If 8 = a tan-1? , then r = a cos 0. Other examples involving a relation between the length of a curve, and quantities contained in the equation to the curve will be given hereafter, when the integrals of more complicated functions have been investigated. 167.] The fundamental relations which exist between the length-element of a curve, its projections on the x- or on the y-axis, and the angle at which either the tangent or the normal at the point (x,y) is inclined to the x-axis, suggest another mode of expressing a curve which is a sufficient defining property of it. Two quantities are of course sufficient to define the plane curve; and I propose to take the length or the length-element, and the angle which the normal makes with the x-axis, for the purpose ; we shall hereby obtain a relation which will express the length of the arc as a function, implicit or explicit, of the angle at which the normals at its extremities are inclined to each other. As this equation is evidently independent of any origin or of any system of coordinates to which the curve may be referred, it has been Damed by Dr. Whewell* the intrinsic equation to the curve. Let dx and dy be the projections of ds, the length-element, on the axes of x and y respectively, and let y be the angle which the normal at (x, y), makes with the x-axis, and let p be the radius of curvature at the point (x, y): then - dx dy = ds; (79) sin y cos y also ds = pdy. (80) • See two Memoirs by Dr. Whewell in vol. VIII, p. 659, and vol. IX. p. 150, of the Cambridge Philosophical Transactions. (81) Now, if by means of these equations, and that of a curve given in terms of x and y, x and y are eliminated, the resulting equation is of the form F(8, y) = 0, and this equation is the intrinsic equation to the curve. If the equation to the curve, of which the intrinsic equation is to be found, is given in polar coordinates, then if $ = tan-1 dy = do+do; (82) and by means of this equation and that to the curve, a relation may be determined in terms of s and y, of the form (81). The process of finding the intrinsic equation by means of the ordinary equation in terms of x and y, or of r and 0, may of course be inverted, and the general equation may be found from the intrinsic equation, if the requisite integrations are possible. 168.] Examples of intrinsic equations. If in (80) we replace p by a, which we will take to be the radius of the circle, s = ay, which is the intrinsic equation to the circle. Ex. 2. The intrinsic equation of the parabola, ya = 4ax. tan y = = (%)*; . .. x = a (tan y)2 dx = 2 a tany (sec y) dy, - dx i = 2a (sec y): dys. sin o Consequently if s begins at the vertex of the parabola, at which point y = 0, Ex. 3. The intrinsic equation of the ellipse. fu a (1-e?) dys 140{1-e2 (sin y)2}t' which result is identical with that given in (37), Art. 158. |