Publisher's Synopsis
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1881 edition. Excerpt: ... parabolic path also renders the action a minimum. When the second fixed point lies on or without the limiting parabola, the discontinuous solution, being the only one which presents itself, undoubtedly renders the action the least possible, as well as a minimum. When both minima are admissible, we shall find that sometimes the one and sometimes the other will give the smaller minimum; and there can be little doubt this smaller minimum is in every case the least possible value also of the action. The comparison of the two minima, when they exist, must be effected by the ordinary calculus, but we subjoin, without proof, the necessary formulas. (See Todhunter's Researches, Art. 173.) Let g be the force of gravity, r, and r, the radii vectores of the two fixed points, C the length of the chord joining these points, and w the action. Then for the parabolic path, according as it subtends less or more than two right angles at the focus, we shall have For the discontinuous solution the action is that due to passing along the verticals only, and is w=: lM(r, + r, ). (20) 221. The principles which have been previously explained regarding the origin and nature of discontinuous solutions arc equally applicable when polar co-ordinates are employed, and we shall find in this case also that they are generally in some manner presented as a solution of the equation M=o, although they may not, and need not always, really satisfy that equation at all. Let us now briefly consider a problem of this kind. Problem XXXVI. // is required to determine whether there be any discontinuous solution involved in Prob. XXII. We have seen, Art. 123, that when the second fixed point lies without a certain limiting ellipse, no elliptic arc, satisfying all the conditions of.
