i.e. according as the space due to the initial velocity, which (see 84. If the path be an ellipse, the period will be 85. If in the preceding cases the direction of the acceleration had been away from the fixed point, the sign of a would have been changed, and the motion would be determined in an exactly similar manner: if we change the sign of λ in the two preceding cases of motion, the equation (b) of Art. 68 would determine the path to be an hyperbola, in which which represents an hyperbola, because by Art. 83, e would then be always > 1. (NEWTON, Props. 11-15). d'u 86. The equation +u= do2 a h2u2 determine the law of variation of the acceleration towards a fixed point, in order that the moving point may trace out a given curve. For a relation is given between u and 0, from which we /d2u and then α = h3u2 +u) gives a, which is usually required in terms of u. E. g. Let the given curve be an ellipse: the acceleration being towards the center. = √ī. (semi-diameter conjugate to r), as before, (Art. 72). For an hyperbola we have only to write-b for b3, and we get a similar result, the only difference being that the sign of is changed, i. e. that the direction of the acceleration is away from the fixed point. 87. Next, let the given curve be a conic section, the acceleration being always towards the focus. where c the semi-latus rectum, and e=the eccentricity; 88. In these and other like cases of motion, the time of describing any part of the path will be found from the equation 2 where 0,0, are the values of at the beginning and end of the time of description that is under consideration, and r is determined in terms of @ from the equation 89. Any other cases of motion that may arise are investigated in a similar manner: the most general equations being where a, a, a, represent the resolved parts of the acceleration in the directions of the co-ordinate axes. The particular artifices to be used for the solution will of course depend on the forms in which a, a, a, appear. 90. In investigating any motion, if we use tangential and normal resolutions, so that our equations of motion are (see instant with changing the magnitude of the velocity without altering its direction; while the other, without causing any instantaneous alteration in its magnitude, changes its direction. This normal acceleration may be taken to measure the tendency to proceed in a straight line. It is sometimes called centrifugal acceleration: the term is not a good one, because the direction of a' is estimated towards the center of curvature, not from it; but whenever this, or any other equivalent term is used, let the 2o is all that is meant. reader know that |