684 MECHANICS Equable descrip tion of areas. Analy tical treat ment of central accelera tion. point O (that of OAB being negative in the second case above), we see that The moment of a diagonal of a parallelogram about any point in the plane of the figure is the (algebraic) sum of the moments of two conterminous sides. Now, suppose the sides of the parallelogram to represent a velocity and its change. If the direction of the change pass through O, its moment is nil. Hence, for acceleration directed towards a fixed point the moment of the velocity about that point is constant. This is commonly expressed by saying that the radius- vector describes equal areas in equal times about the point to which the acceleration is directed. For the moment of the velocity is double the area so traced in unit of time. Another way of expressing the same thing is to say that the angular velocity of the radius-vector is inversely as the square of its length. For the product of the square of the radius-vector and its angular velocity is double the area described by it in unit of time. The converse of this proposition is also evidently true ; i.e., when a point moves so that the moment of the velocity about a point in the plane of its motion is constant, its acceleration relative to that point (if any) is directed towards or from that point. 47. Analytically : if P be the acceleration, directed towards a fixed point which we choose as origin, we have x = -Pcos0 = -Px/r, y = - Psin0 = -Py/r, (r and 6 being the polar coordinates of the moving point ; we have already seen that the path is necessarily plane). Eliminating P, we have Thus r z d const. = h . Polar coordi nates. This may be transformed, at once, by the methods of the differ ential calculus, into ps=pv = 7i i where p is the length of the perpendicular from the origin to the tangent to the path. Conversely, if equal areas be described by the radius-vector in equal times, we have r^6 = xy yx = Ti . Whence xy-yx=Q, or x = Qx, y=Q,y. Hence the whole acceleration is Q,r, and is directed t owards or from the origin. While we are dealing with these formulae we may investigate the general expressions for velocity and acceleration in terms of polar coordinates for a point moving in a plane. We have x=rcosd, y = rsin0. From these Hence the speed along the radius-vector is x cos 6 + y sin 6 r ; and that perpendicular to the radius-vector (in the direction in which Q increases) is These expressions might have been written down at once, if we note that Sr and rSQ are the resolved parts of 8s along, and perpen dicular to, r. But we must be careful how we carry this species of reasoning one step further. Taking the second fluxions of x and y, we have x = (f - rd") cos - (If 6 + r6) sin 0, y=(f- rd 2 ) sin + (2ff) + rB) cos . Hence the acceleration along the radius-vector is and that perpendicular to it (positive when in the direction in which 6 increases) is
- cos 6 - x
Thus, although r represents truly the speed along r, r does not represent the acceleration in that direction. It represents, in fact, only the acceleration of speed along r. But we have seen that there is acceleration along r, if its direction changes, even when its length is constant, i. e. , when the path is circular ; and in that case rfr 2 is the quantity which we designated as pco 2 in 38. As a verification of these formulas, let us consider uniform motion in a straight line. Here the equation of the straight line, and the condition of uniform motion. We have j* = a sec tan 0.0 , V = asee 2 0J = r 2 #/a; f = Vsiufl, we have Here, although there is no acceleration, f has a definite value. But r r(fi = a 2 V 2 /V 3 o? V 8 /?* 3 = - From the expressions for the acceleration along and perpendicular to the radius-vector we at once obtain the result above ( 46). For, if there be no acceleration perpendicular to the radius-vector, -(f*)-o r dt from which r s = const. =li. We have, in addition to this, the expression for the acceleration towards the origin, r-r&*= -P. Eliminating 6, we have This gives r in terms of t, and thus reduces (if we please) any case Eeduc of a central orbit to a corresponding case of rectilinear motion, tionto The difference between the accelerations in the revolving radius- case oi vector and in the fixed line is a term depending on the inverse recti- cube of the radius-vector. But the usual mode of proceeding is as linear follows. motioi Multiply by fdt and integrate, then d$ = C - 2/Pdr. The left-hand member obviously represents the square of the velocity, as it is the sum of the squares of f and r6. For we have . dr , drh This gives a relation between r and 0, which is therefore the polar equation of the path described. It is usual to employ, instead of r, its reciprocal l/r = u. With this the equation be- "Pdu Differentiating with regard to 0, and dividing by 2& 2 , we ob tain finally d0 do 2 W-u? an equation of very great importance. When there is acceleration T perpendicular to the radius-vector, as well as - P along it, this equation takes the form Polar equati of pat d8 2 48. There are two specially important cases of central Specia acceleration. The first is that of the gravitation law, cases - the other that of Hooke s law. We will take these in order, but by very different methods. 49. Planetary Motion. With the gravitation law the Plane- acceleration varies inversely as the square of the distance tary from the point to which it is directed. But, as we have motl01 just seen, the angular velocity of the radius-vector, i.e., of the direction of acceleration, varies according to the same law. Hence in the hodograph, the linear velocity (whose
magnitude is that of the acceleration in the path) is pro-