MECHANICS where I and e are two new arbitrary constants introduced in place of a and H. With these (5) becomes i- -/-r= J r Vfe 2 - = 0- l)/l 2 +2/lr-l/r* -= = 0- cos- 1 (r- a -Z- ] ) , ma <? x + l+ccos(6-a 1 ) the general polar equation of conic sections referred to the focus. Also by differentiating (5) with respect to r, we have adr . = dO , r a V2(H+ju/r)-a 2 /r a from which, by (6), we immediately obtain 1 /- 1 /" F = V r s d0 = T= J r ie . This involves, again, the equation of equable description of areas. Compare 144. Action in 210. In a planet s elliptic orbit the time is measured by the area orbit of described about one focus, and the action by that described about the planet, other. For with the usual notation we have h ~~P (S by the result of 47. But in the ellipse or hyperbola, p being the perpendicular from the second focus, Hence p ds , which expresses the result stated above. It is easy to extend this to a parabolic orbit, for which, indeed, the theorem is even more simple. Ex- 211. Unit particles are projected simultaneously and horizontally ample, in all directions from every point of a vertical axis, all having the same total energy at starting; find the surfaces of equal action. "We may obviously confine ourselves to a plane section through the axis. Let x be the vertical coordinate of a particle, measured downwards from the level at which the common energy is wholly potential, k the coordinate of the point in the axis from which it was projected. Then we have, after the lapse of time t, Eliminating t, we have the equation of the parabolic path To find the orthogonal trajectory (the meridian section of the surface of equal action), differentiate, put - for - , and eliminate k. We dy dx thus have dy dx JL 2k y /x + y - /x - 1 y x + /x- - y 2 Vic + 2/ + lx - y or - /x + y(dx + dy) + ^/x-y(dx-dy) = Q , so that . (x + yfi (x - y}$ = const. If we turn the axes through {w in their own plane, the coordinates being now { and 77, ^ this equation becomes , |i + r) ? = a ?. B In fig. 55, AM is the axis. A few of the paths are shown by full lines, and two of the sections of sur- G faces of equal action ^ by dotted lines. These sections indicate cusps lying on the line AH, which M makes an angle |TT with the vertical, and is touched by all the paths. The path whose vertex is G touches this line in IT, and therefore passes through the cusp of which the branches are HK and HL. HK belongs to all paths whose vertices are above G, HL to those (such as ML) whose vertices are below G. It is worthy of note that, by the first equations above, 725 by the substitution of which in the equation of action we see how the time of reaching a particular surface of equal action depends upon the position of the starting point. 212. A very interesting plane example, which has elegant Plane applications in fluid motion, and in the conduction of electric currents currents in plates of uniform thickness, is furnished by assuming A = logr, or A = 0, where r and are the polar coordinates of the moving particle. In the former, where the curves of equal action are circles with the origin as centre, we have dA_ = . _x_ dA_ y dx ~r 2 ify~*"V so that the paths are radii vectores described with velocity 1/r, Also we have so that the force is central, and its value is In the second case, where the curves of equal action are radii drawn from the pole and we have dA_ = . = _ y_ dA . _x^ dx~ r* dy~ U ~~r* The kinetic energy is still l/2r 2 , and the central force- 1/r 3 , but the paths are circles with the origin as centre. Thus the lines of equal action and the paths of individual particles are convertible. We have also, in each of these cases, dx) dij t This shows (as in 94) that, whatever be originally the grouping of a set of particles moving all according to one or other of these conditions, the density at any part of the group remains unchanged during the motion. In fact, as it is easy to prove, A and A are elementary solutions of the partial differential equation _ dx 2 * dy* and they are conjugate, in the sense that (i); dx dx dy dy For this reason the paths belonging to the two systems are every where orthogonal to one another. Also, as the differential equation (1) for A is linear, any linear function of particular integrals is an integral. Thus, for instance, we may take (p being any constant) with A *=p log r + 6 . These, representing orthogonal sets of logarithmic spirals, possess the same properties with regard to action as did the concentric circles and their radii, which, in fact, are the mere particular case when p = 0. 213. It is easy to give graphic methods of tracing these curves Graphic of action by means of an old process recently much developed by method. Clerk Maxwell. The present example, though a very simple one, is quite sufficient to illustrate the process. Draw, as in fig. 56, a set of circles whose radii are 6, 6 ", g 3 ", &c., and a set of radii vectores making with the initial line the successive angles, 2 3 , &c. , a being a quantity which P P P may have any convenient value. These lines will form a network, finer as a is smaller. Now suppose we wish to trace the curve A= na. We take the intersection of the circle whose radius is "* with the radius- vector corresponding to the angle (.-). Thus we have for the value of A at the point of intersection P --no. , as required. By marking the intersections corresponding to different values of s,
we have a series of points in the required curve which, by adjust-