It may be remarked that equations (3) are of the same form as the equations for finding the resultant of a set of forces acting at a point in one plane. 43. Compositions of two S.H. motions of the same period at right angles, differing by a quarter-period in epoch. If the equation of one of them be that of the other will be x=a cos 0, y=b cos (6±7) = 6 sin 9. Hence (25) (~~)2 + (31⁄2)2 = cos2 0 + sin20= 1, the equation to an ellipse of semi-axes a b. (5) 44. Composition of any two S.H. motions of the same This is an equation of the second degree, and as we know that the resultant motion must be in some closed curve, the locus is, in general, an ellipse. Describe a rectangle whose sides are the double amplitudes of the two given components. Then all the ellipses obtained by giving different values to can be inscribed in this rectangle, since the extreme values of x are always ±a, and the extreme values of y are ±6. It will be found upon examination that the direction in which the moving point travels round the ellipse depends upon the value of ò, and is reversed as d passes through the values o and π. We shall examine the two 62 article the same equation y2= (a2-x2). a2 Hence in one quarter period the motion is from the point x=a, y=o to the point x=0, y=b. The direction of revolution is therefore from the positive axis of x to the positive axis of y'. Hence the direction of revolution is from the positive axis of x to the negative axis of y; which is opposite to the direction of revolution in the preceding case. 45. Next, let us suppose the periods of the two mutually perpendicular components to be only approximately equal. Then the resultant motion at any moment will be approximately one of the ellipses represented by equation (7), but will gradually change, and thus, instead of the motion repeating itself in a fixed ellipse, it will approximate in succession to all the ellipses which equation (7) can be made to represent by giving every possible value to ò. The first figure in Plate III. shows the trace left, by a point describing these approximate ellipses, upon a sheet of paper travelling uniformly past it. If d is increasing, the x vibrations are gaining upon the y vibrations. For the phase of the x vibrations is 6, and the phase of the y vibrations is -; but if d increases with (and at a much slower rate) the increment of 6-8 in any time is less than the increment of 0; that is to say, the phase of the y vibrations increases more slowly than the phase of the x vibrations. The opposite will be the case if & is decreasing. 45.* To investigate the resultant of two S.H. motions along lines inclined at an angle other than a right angle, we have only to suppose the axes of co-ordinates in the preceding sections 43-45 to be oblique. The algebraic work is unchanged, and the interpretation of the results presents no difficulty. It may be remarked that equations (3) are of the same form as the equations for finding the resultant of a set of forces acting at a point in one plane. 43. Compositions of two s.H. motions of the same period at right angles, differing by a quarter-period in epoch. If the equation of one of them be the equation to an ellipse of semi-axes a b. 44. Composition of any two S.H. motions of the same period, at right angles. Their equations may be written x=a cos 4, A y=b cos (0-6)=6 (cos e cos + sin sin 6). This is an equation of the second degree, and as we |