Notation for con courses, and 359. In such statements as this it will frequently be convenient to indicate particular configurations of the system by figurations, single letters, as O, P, Q, R; and any particular course, in action. which it moves through configurations thus indicated, will be called the course O...P...Q...R. The action in any natural course will be denoted simply by the terminal letters, taken in the order of the motion. Thus OR will denote the action from O to R; and therefore OR = — RO. When there are more real natural courses from 0 to R than one, the analytical expression for OR will have more than one real value; and it may be necessary to specify for which of these courses the action is reckoned. Thus we may have == OR for O...E...R, OR for O...E...R, OR for O...E"...R, three different values of one algebraic irrational expression. minimum 360. In terms of this notation the preceding statement Theorem of (§ 358) may be expressed thus:-If, for a conservative system, action. moving on a certain course O...P... O'...P', the first kinetic focus conjugate to O be O', the action OP', in this course, will be less than the action along any other course deviating infinitely little from it: but, on the other hand, OP' is greater than the actions in some courses from 0 to P' deviating infinitely little from the specified natural course O...P...O'...P'. courses of 361. It must not be supposed that the action along OP is Two or more necessarily the least possible from 0 to P. There are, in fact, minimum cases in which the action ceases to be least of all possible, before possible, E action Ο E' E F Two or more a kinetic focus is reached. courses of action minimum, minimum, geodetic lines be points. Thus if OEAPO'E'A' be a sinuous minimum geodetic line cutting the outer circle of an anchor-ring, or possible.. the equator of an oblate spheroid, in successive points 0, A, A', it is easily seen that O', the first kinetic focus conjugate to O, must lie somewhat beyond A. But the length OEAP, although a minimum (a stable position for a Case of two stretched string), is not the shortest distance on the surface and one not from 0 to P, as this must obviously be a line lying entirely on one side of the great circle. From 0, to any point, Q, short of tween two A, the distance along the geodetic OEQA is clearly the least possible: but if Q be near enough to A (that is to say, between A and the point in which the envelope of the geodetics drawn from O, cuts OEA), there will also be two other geodetics from O to Q. The length of one of these will be a minimum, and that of the other not a minimum. If Q is moved forward to A, the former becomes OE A, equal and similar to OEA, but on the other side of the great circle: and the latter becomes the great circle from 0 to A. If now Q be moved on, to P, beyond A, the minimum geodetic OEAP ceases to be the less of the two minimums, and the geodetic OFP lying altogether on the other side of the great circle becomes the least possible line from 0 to P. But until P is advanced beyond the point, O', in which it is cut by another geodetic from O lying infinitely nearly along it, the length OEAP remains a minimum, according to the general proposition of § 358, which we now proceed to prove. Difference between two sides and the third of a kinetic triangle. (a) Referring to the notation of § 360, let P, be any configuration differing infinitely little from P, but not on the course O...P...O'...P'; and let S be a configuration on this course, reached at some finite time after P is passed. Let 4, 4,... be the co-ordinates of P, and 4, 4,,... those of P,, and let But if έ, ŋ,... denote the components of momentum at P in the Difference course 0...P, which are the same as those at P in the continua- sides and tion, P....S, of this course, we have [§ 330 (18)] between two the third of a kinetic triangle. Hence the coefficients of the terms of the first degree of d4, 84, (d2 (OP+PS) ď2 (OP + PS) 8¥8$ + OP + P ̧S - OS = }} dy (b) Now, assuming 842+2 (1). according to the known method of linear transformations, let the whole number of degrees of freedom being i This may be done in an infinite variety of ways; and, towards fixing upon one particular way, we may take a = 4, ß. = 4, etc.; and subject the others to the conditions ψα + φβ + ... = 0, ψα + φβ This will make A, 1 = O, etc. 0: for if for a moment we suppose P, to be on the course 0...P...O', we have But in this case OP + PS = OS; and therefore the value of the where R denotes a remainder consisting of terms of the third Difference between two sides and the third of a kinetic triangle. (c) Another form, which will be used below, may be given to the same expression thus:-Let (,, n,, .,...) and (,', n', ',...) be the components of momentum at P,, in the courses OP and PS respectively. By § 330 (18) we have OP + PS - OS = − 1 {(§' — §') d↓ + (n,' − n,) d$ + ... } }) + R (5), where R denotes a remainder consisting of terms of the third and higher degrees. Also the transformation from du, dp, ... to ,,,..., gives clearly (d) Now for any infinitely small time the velocities remain sensibly constant; as also do the coefficients (4, 4), (4, 4), etc., in the expression [§ 313 (2)] for T: and therefore for the action we have - √2T {(4, 4) (4 − 4 ̧)3 + 2 (4, 4) (↓ − 4) (4 − ¢ ̧) + etc.} where (...) are the co-ordinates of the configuration from which the action is reckoned. Hence, if P, P', P' be any three configurations infinitely near one another, and if Q, with the proper differences of co-ordinates written after it, be used to denote square roots of quadratic functions such as that in the preceding expression, we have PP' =√2T.Q{(4 - 4'), ($ - $′), ...}) ·(7). In the particular case of a single free particle, these expressions become simply proportional to the distances PP′, P'P", P′′P; and by Euclid we have PP+PP" <P'P" unless P is in the straight line P'P". The verification of this proposition by the preceding expressions (7) is merely its proof by co-ordinate geometry with an oblique rectilineal system of co-ordinates, and is necessarily somewhat complicated. If (4, 6) = (6, 0)=(0, 4) = 0, the co-ordinates become rectangular and the algebraic proof is easy. There is no difficulty, by following the analogies of these known processes, to prove that, for any number of co-ordinates, 4, 4, etc., we have PP+PP"> P'P Difference between two sides and the third of a kinetic triangle. (expressing that P is on the course from P to P'), in which case P'P+ PP" = PP", P'P, etc., being given by (7). And further, by the aid of (1), it is easy to find the proper expression for P'P+PP" - P'P", when P is infinitely little off the course from P' to P"; but it is quite unnecessary for us here to enter on such purely algebraic investigations. (e) It is obvious indeed, as has been already said (§ 358), that the action along any natural course is the least possible between its terminal configurations if only a sufficiently short course is included. Hence for all cases in which the time from 0 to S is less than some particular amount, the quadratic term in the expression (3) for OP,+P,S-OS is necessarily positive, for all values of x,, x,, etc.; and therefore A,, A,,...A,, must each be positive. (f) Let now S be removed further and further from O, along the definite course 0...P...O', until it becomes O'. O', let P, be taken on a natural course through O VOL. I. Actions on courses in different When it is finitely near and O', de one another 28 |