688 MECHANICS Simple harmonic motions at right angles. Periods nearly equal. Simple harmonic motions of one period, in any direc tions. phases of the motions in OB and OA. Then if P, Q represent at time t the corresponding positions in the common circle, we have arc FQ = arc EP ; and if perpen diculars be drawn, PM to OA, and QN" to OB, their intersection S is the position at time t in the resultant motion. The locus of S is, by what has been proved above, an ellipse which touches the sides of the square CDC D . When EOF is a right angle, i.e., when the phases are alike, this ellipse becomes ,the diagonal CC of the square touching the circle at the extremities of AA and BB . When EOF is three right angles, the ellipse becomes the diagonal DD . When it is two right angles, or four, i.e., when OB is one quarter, or three quarters, of a period in advance of OA, the ellipse becomes the circle ABA B . To find in any case whether it is described positively or negatively ( 44), we have only to notice how OS turns. Now while P is near A, MS remains closely coincident with AC. If, then, Q be anywhere in the semicircle BA B , N moves in the direction BB and the angle AOS diminishes. Hence the ellipse is described negatively (or in the direction of the hands of a watch) if the epoch of the motion in OB exceeds that of the motion in OA by anything up to two right angles. And similar reasoning shows that, if the excess be from two to four right angles, the ellipse is described positively. If the amplitudes be not equal, we have only to extend or foreshorten the figure parallel to OA or to OB. The square CDC D becomes a rectangle, in which the orbits (all of which, with the exception of the diagonals, are now ellipses) are inscribed. Everything else is as before, 61. When the periods in the two component motions are nearly, but not quite, equal, the phase of one gains gradually on the other, and the path passes continuously through the forms of all the possible ellipses, but remains possessed of the one property common to them all. It becomes a species of spiral, but in every convolution it touches, in succession, each side of the square or rectangle above discussed. 62. Similar reasoning shows that the superposition of any number of simple harmonic motions in any directions and with any amplitudes and differences of phase, provided the period is the same for all, gives rise to motion in an ellipse about the centre. But this follows more easily from analysis. Take, first, two simple harmonic motions of the same period parallel to the axes of x and y. We have e ). Eliminating t between these equations, we have at once z 2 9 , / x m 2 (e -e), the equation of an ellipse. It becomes a circle when and only when a = a , cos(e ~ e) = 0, i.e., when the amplitudes are equal, and the phases differ by an odd number of right angles. It becomes the straight line x/a - y/a = , when e - e is zero ; and x/a + y/a = , when e - e is two right angles. If SOA be called 0, we have = (cos(e - e) - sin(e - Hence, taking the fluxion of each side, . 6= -- wsin(e - ct Thus, as before, 6 is essentially negative, i.e., the rotation in the ellipse is right-handed if e - e lie between and ir, left-handed if it lie between ir and 2?r. For a simple harmonic motion, denoted by = cos(cu+ e), in a line whose direction cosines are I, m, n, we have the com ponents l, ?, nl- parallel to the three axes respectively. Hence for the resultant of any number of such, all having the same period, we have x=~2, . alcos(ut + e) = cos wt2,(al cost)- sin ut2(al sine). Thus we have three equations of the form x=A.coscat- A sinut , y = B cos ut - B sin ut , z = C cos ut C sin cat . If we take three quantities A, p., y, such that we have also The first two equations determine without ambiguity the ratios of fj. and v to A. Hence the third is the equation of a definite plane in which the path lies. We may now choose this plane as that of x, y. The value of z above, becomes identically zero ; and the elimination of t between the equations for x and y gives the ellipse as before. 63. When the periods of the simple harmonic motions Period are not equal we have not , equal. x=acos(<ot It is easy to trace the corresponding curve by points ; but, except when there is a simple numerical ratio between co and to , the equation cannot be presented as an algebraic one between x and y. If 2to = co, we may shift the epoch so that the equations may be written Eliminating t from the first, by the help of the second, we have Zy -- This denotes, in general, a curve of the fourth order, of a figure-of- 8 form, as in fig. 19. When a = ir the curve is a portion of a Fig. 19. parabola, its vertex being to the right or left as n is odd or even. This parabola corresponds, in the present case, to the straight lines in the casa of 62. When the periods differ slightly from the ratio 2:1, the path passes in succession through the forms traced, for ward and backward alternately ; and, each time that it opens out from the parabolic form, the tracing-point describes it in the opposite direction to that in which it described it before the path collapsed into the parabola. 64. The principles already illustrated are sufficient Com- for the examination of every case of this kind. But one positic or two particular cases merit special notice. The case of of _ .? . .. r ,. T .,. umfor; two uniform circular motions of equal periods, in one c i rcu i a plane, we have already noticed ( 54). Q describes its motioi circle about P, P its circle about 0, and the result is uniform circular motion of Q about 0. The radius of this circle may be equal to the sum or difference of the radii of the separate circles, or may have any intermediate value, according to the difference of phase. If the periods be not exactly equal, the motion takes place virtually in a circle whose radius continuously oscillates between the above limits. The path is a species of spiral, which lies between two concentric circles of these radii. 65. When the component circular motions are in opposite directions, we have an extremely interesting and important case. It is obvious that there must now be
positions in which OP and PQ are in the same straight