for con

courses, and

359. In such statements as this it will frequently be con- Notation venient 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,

minimum

three different values of one algebraic irrational expression. 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...0'...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, minin:um cases in which the action ceases to be least of all possible, before possible.

E

action

action

minimum,

minimum,

geodetic

lines be

points.

Thus if OEAPO'E'A' be a sinuous
outer circle of an anchor-ring, or
spheroid, in successive points 0,
that O', the
that O', the first kinetic focus
somewhat beyond A. But the

possible. the equator of an oblate A, A', it is easily seen conjugate to O, must lie 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 O, 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 0, 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 OEA, 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 0...P...O'...P'; and let S be a configuration on this course, reached at some finite time after P is passed. Let v, p,... be the co-ordinates of P, and 4,, ò̟,,....... those of P,, and let

But if §, î,....... denote the components of momentum at P in the Difference course O...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 dy, 84,
in the preceding expression vanish, and we have

according to the known method of linear transformations, let

α

a

a1, B1,... α, B.,... be so chosen that the preceding quadratic function be reduced to the form

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 = †, ß¡= $, etc.; and subject the others to the conditions

ψα, + β + ... = 0, Ψα + β + = 0, etc.

1

2

This will make 4,0: for if for a moment we suppose P, to be on the course 0...P...O', we have

But in this case OP,+P,S=OS; and therefore the value of the quadratic must be zero; that is to say, we must have A.= 0. Hence we have

OP,+P,S—OS= }} (4,‚μ‚2 + A‚¤‚ ̧2 + ... + Å¡_‚o¡_

2

Ax

2 2

(3)

where R denotes a remainder consisting of terms of the third and higher degrees in dy, dp, etc., or in x, x,, etc.

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

and so for ŋ,' —~,, etc. Hence (1) is the same as

OP, + P,S — OS = − } { (§,' — §') 84 + (n,' — n,) dp + ... } }

+ R

(5),

where R denotes a remainder consisting of terms of the third and higher degrees. Also the transformation from du, dp, ... to X1, X2, ..., gives clearly

27

(d) Now for any infinitely small time the velocities remain sensibly constant; as also do the coefficients (4, 4), (4, 6), etc., in the expression [§ 313 (2)] for T: and therefore for the action we have

f2T'dt=√2Tƒ√2Tdt

· √2T {(4, 4) (4 − 4。)2 + 2 (4, 4) (4 − 4.) (4 − 4.) + etc.}} where (4%, Þ。, ...) 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

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

PPPP" <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) = (p, 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, y, p, etc., we have P'P + PP" > P'P",

(expressing that P is on the course from P to P''), in which case P'P+PP" = P'P",

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 X1, X27 etc.; and therefore A ̧, A ̧‚.....A¡_, must each be positive.

i-1

Difference between two sides and the third of

a kinetic triangle.

different

(ƒ) Let now S be removed further and further from O, along Actions on the definite course 0...P...O', until it becomes O'. O', let P, be taken on a natural course through O

VOL. I.

When it is courses inand O', de- finitely near

28

one another