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reference to which the doctrine comes into question, into two classes.
Class L Those in which the velocities considered are either constant or only vary suddenly—-that is to say, in infinitely small times—or in times so short that they may be omitted from the time-integration. To this class belong:
(a) The original Waterston-Maxwell case and the collisions of ideal rigid bodies of any shape, according to the assumed law that the translatory and rotatory motions lose no energy in the collisions.
(b) The frictionless motion of one or more particles constrained to remain on a surface of any shape, this surface being either closed (commonly called finite though really endless), or being a finite area of plane or curved surface, bounded like a billiard-table, by a wall or walls, from which impinging particles are reflected at angles equal to the angles of incidence.
(c) A closed surface, with non-vibratory particles moving within it freely except during impacts of particles against one another or against the bounding surface.
(d) Cases such as (a), (b), or (<?), with impacts against boundaries and mutual impacts between particles, softened by the supposition of finite forces during the impacts, with only the condition that the durations of the impacts are so short as to be practically negligible in comparison with the durations of free paths.
Class II. Cases in which the velocities of some of the particles concerned sometimes vary gradually ; so gradually that the times during which they vary must be included in the time-integration. To this class belong examples such as (d) of Class I. with durations of impacts not negligible in the time-integration.
§ 29. Consider first Class I. (b) with a finite closed surface as the field of motion and a single particle moving on it. If a particle is given, moving in any direction through any point I of the field, it will go on for ever along one determinate geodetic line. The question that first occurs is, Does the motion fulfil Maxwell’s condition (see § 18 above) ? that is to say, for this case, If we go along the geodetic line long enough, shall we pass infinitely nearly to any point Q whatever, including I, of the surface an infinitely great number of times in all directions? This question cannot be answered in the affirmative without reservation. For example, if the surface be exactly an ellipsoid it must be answered in the negative, as is proved in the following §§ 30, 31, 32.
§ 30. Let, A A', BB', CC'> be the ends of the greatest, moan, and least diameters of an ellipsoid. Let Uj U2 U3 U4 be the umbilics in the arcs AC, CA', A'O', C'A. A known theorem in the geometry of the ellipsoid tells us, that every geodetic through Ux passes through XJ3? and every geodetic through U2 passes through U4. ^This statement regarding geodetic lines on an ellipsoid of three unequal axes is illustrated by fig. 1, a diagram showing for the extreme case in which the shortest axis is zero, the exact construction of a geodetic through JJ1 which is a focus of the ellipse shown in the diagram. U3, Cr, U4 being infinitely near to U2, 0, XJi respectively are indicated by double letters at the same points. Starting from Ui draw' the geodetic UiQUs ; the two parts
of which UiQ and QU3 are straight lilies. It is interesting to remark that, in whatever direction we start from TTi, if we continue the geodetic through U3, and on through again and so on endlessly, as indicated in the diagram by the straight lines U1QU3Q,U1Q/,U3Q//,7 and so on, we come very quickly to lines approaching successively more and more nearly to coincidence with the major axis. At every point where the path strikes the ellipse it is reflected at equal angles to the tangent. The construction is most easily made by making the angle between the reflected path and a line to one focus, equal to the angle between the incident path and a line to the other focus.
§ 31- Returning now to the ellipsoid :—From any point I, between Ul and U2? draw the geodetic IQ, and produce it