of the great circle. From 0 to any point, Q, short of 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 envelop of the geodetics drawn from 0, cuts OEA), there will also be two other geodetics from 0 to Q. The length of one of these will be a minimum, and that of the other not a minimum. If 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 be moved on to P, beyond A, the minimum geodetic OEAP ceases to be the less of the two minima, and the geodetic OFP lying altogether on the other side of the great circle becomes the least possible line from O 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 § 311. 315. As it has been proved that the action from any configuration ceases to be a minimum at the first conjugate kinetic focus, we see immediately that if Ø be the first kinetic focus conjugate to O, reached after passing O, no two configurations on this course from 0 to can be kinetic foci to one another. For, the action from O just ceasing to be a minimum when O' is reached, the action between any two intermediate configurations of the same course is necessarily a minimum. 316. When there are ¿ degrees of freedom to move there are in general, on any natural course from any particular configuration, O, at least - 1 kinetic foci conjugate to O. Thus, for example, on the course of a ray of light emanating from a luminous point O, and passing through the centre of a convex lens held obliquely to its path, there are two kinetic foci conjugate to O, as defined above, being the points in which the line of the central ray is cut by the so-called 'focal lines'1 of a pencil of rays diverging from O and made convergent after passing through the lens. But some or all of these kinetic foci may be on the course previous to 0; as, for instance, in the case of a common projectile when its course passes obliquely downwards through O. Or some or all may be lost, as when, in the optical illustration just referred to, the lens is only strong enough to produce convergence in one of the principal planes, or too weak to produce convergence in either. Thus also in the case of the undisturbed rectilineal motion of a point, or in the motion of a point uninfluenced by force, on an anticlastic surface (309), there are no real kinetic foci. In the motion of a projectile (not confined to one vertical plane) there can be only one kinetic focus on each path, conjugate to one given point; though there are three degrees of freedom. Again, there may be any number 1 In our second volume we hope to give all necessary elementary explanations on this subject. more than i-1 of foci in one course, all conjugate to one configuration, as for instance on the course of a particle, uninfluenced by force moving round the surface of an anchor-ring, along either the outer great circle, or along a sinuous geodetic such as we have considered in § 311, in which clearly there are an infinite number of foci each conjugate to any one point of the path, at equal successive distances from one another. 317. If i-1 distinct1 courses from a configuration O, each differing infinitely little from a certain natural course O.. E.. O1 . . 02. Oi-..Q, cut it in configurations O1, O2, O3,... Oi-1, and if, besides these, there are not on it any other kinetic foci conjugate to 0, between 0 and Q, and no focus at all, conjugate to E, between E and Q, the action in this natural course from 0 to Q is the maximum for all courses O...P,, P, . . . Q; P, being a configuration infinitely nearly agreeing with some configuration between E and O1 of the standard course O. . E . . O1.. 02.... Oi-1.. Q2 and O... P,, P, . . . Q denoting the natural courses between 0 and P,, and P, and Q, which deviate infinitely little from this standard course. 1 318. Considering now, for simplicity, only cases in which there are but two degrees (§ 165) of freedom to move, we see that after any infinitely small conservative disturbance of a system in passing through a certain configuration, the system will first again pass through a configuration of the undisturbed course, at the first configuration of the latter at which the action in the undisturbed motion ceases to be a minimum. For instance, in the case of a particle, confined to a surface, and subject to any conservative system of force, an infinitely small conservative disturbance of its motion through any point, O, produces a disturbed path, which cuts the undisturbed path at the first point, Ơ, at which the action in the undisturbed path from O ceases to be a minimum. Or, if projectiles, under the influence of gravity alone, be thrown from one point, O, in all directions with equal velocities, in one vertical plane, their paths, as is easily proved, intersect one another consecutively in a parabola, of which the focus is O, and the vertex the point reached by the particle projected directly upwards. The actual course of each particle from O is the course of least possible action to any point, P, reached before the enveloping parabola, but is not a course of minimum action to any point, Q, in its path after the envelop is passed. 319. Or again, if a particle slides round along the greatest circle of the smooth inner surface of a hollow anchor-ring, the action,' or simply the length of path, from point to point, will be least possible for lengths (§ 305) less than Tab. Thus if a string be tied round outside on the greatest circle of a perfectly smooth anchor-ring, it will slip off unless held in position by staples, or checks of some kind, at 1 Two courses are here called not distinct if they differ from one another only in the absolute magnitude, not in the proportions, of the components of the deviations by which they differ from the standard course. distances of not less than this amount, π√ab, from one another in▾ succession round the circle. With reference to this example, see also § 314, above. Or, if a particle slides down an inclined hollow cylinder, the action from any point will be the least possible along the straight path to any other point reached in a time less than that of the vibration one way of a simple pendulum of length equal to the radius of the cylinder, and influenced by a force equal to g cos i, instead of g the whole force of gravity. But the action will not be a minimum from any point, along the straight path, to any other point reached in a longer time than this. The case in which the groove is horizontal (i=0) and the particle is projected along it, is particularly simple and instructive, and may be worked out in detail with great ease, without assuming any of the general theorems regarding action. CHAPTER III. EXPERIENCE. 320. By the term Experience, in physical science, we designate, according to a suggestion of Herschel's, our means of becoming acquainted with the material universe and the laws which regulate it. In general the actions which we see ever taking place around us are complex, or due to the simultaneous action of many causes. When, as in astronomy, we endeavour to ascertain these causes by simply watching their effects, we observe; when, as in our laboratories, we interfere arbitrarily with the causes or circumstances of a phenomenon, we are said to experiment. 321. For instance, supposing that we are possessed of instrumental means of measuring time and angles, we may trace out by successive observations the relative position of the sun and earth at different instants; and (the method is not susceptible of any accuracy, but is alluded to here only for the sake of illustration) from the variations in the apparent diameter of the former we may calculate the ratios of our distances from it at those instants. We have thus a set of observations involving time, angular position with reference to the sun, and ratios of distances from it; sufficient (if numerous enough) to enable us to discover the laws which connect the variations of these co-ordinates. Similar methods may be imagined as applicable to the motion of any planet about the sun, of a satellite about its primary, or of one star about another in a binary group. 322. In general all the data of Astronomy are determined in this way, and the same may be said of such subjects as Tides and Meteorology. Isothermal Lines, Lines of Equal Dip or Intensity, Lines of No Declination, the Connexion of Solar Spots with Terrestrial Magnetism, and a host of other data and phenomena, to be explained under the proper heads in the course of the work, are thus deducible from Observation merely. In these cases the apparatus for the gigantic experiments is found ready arranged in Nature, and all that the philosopher has to do is to watch and measure their progress to its last details. 323. Even in the instance we have chosen above, that of the planetary motions, the observed effects are complex; because, unless possibly in the case of a double star, we have no instance of the undisturbed action of one heavenly body on another; but to a first approximation the motion of a planet about the sun is found to be the same as if no other bodies than these two existed; and the approximation is sufficient to indicate the probable law of mutual action, whose full confirmation is obtained when, its truth being assumed, the disturbing effects thus calculated are allowed for, and found to account completely for the observed deviations from the consequences of the first supposition. This may serve to give an idea of the mode of obtaining the laws of phenomena, which can only be observed in a complex form; and the method can always be directly applied when one cause is known to be pre-eminent. 324. Let us take a case of the other kind-that in which the effects are so complex that we cannot deduce the causes from the observation of combinations arranged in Nature, but must endeavour to form for ourselves other combinations which may enable us to study the effects of each cause separately, or at least with only slight modification from the interference of other causes. A stone, when dropped, falls to the ground; a brick and a boulder, if dropped from the top of a cliff at the same moment, fall side by side, and reach the ground together. But a brick and a slate do not; and while the former falls in a nearly vertical direction, the latter describes a most complex path. A sheet of paper or a fragment of gold-leaf presents even greater irregularities than the slate. But by a slight modification of the circumstances, we gain a considerable insight into the nature of the question. The paper and gold-leaf, if rolled into balls, fall nearly in a vertical line. Here, then, there are evidently at least two causes at work, one which tends to make all bodies fall, and that vertically; and another which depends on the form and substance of the body, and tends to retard its fall and alter its vertical direction. How can we study the effects of the former on all bodies without sensible complication from the latter? The effects of Wind, etc., at once point out what the latter cause is, the air (whose existence we may indeed suppose to have been discovered by such effects); and to study the nature of the action of the former it is necessary to get rid of the complications arising from the presence of air. Hence the necessity for Experiment. By means of an apparatus to be afterwards described, we remove the greater part of the air from the interior of a vessel, and in that we try again our experiments on the fall of bodies; and now a general law, simple in the extreme, though most important in its consequences, is at once apparent-viz. that all bodies, of whatever size, shape, or material, if dropped side by side at the same instant, fall side by side in a space void of air. Before experiment had thus separated the phenomena, hasty philosophers had rushed to the conclusion that some bodies possess the quality of heaviness, others that of lightness, etc. Had |