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§ 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
through Q on the ellipsoidal surface. It must cut the arc A'C'A at some point between U3 and U4, arid, if continued on and on, it must cut the ellipse ACA'O'A successively between Ui and U3, or between U3 and U4 ; never between U2 and U3, or U4 and Uj. This, for the extreme case of the smallest axis zero, is illustrated by the path IQQ'Q^Q"' Q,VQV in fig. 2.
§ 32. If now, on the other hand, we commence a geodetic through any point J betw een Ui and U4, or between U2 and 1J3, it will never cut the principal section containing the umbilicus, either between U, and U2 or between U3 and U4. This, for the extreme case of CC^O, is illustrated in fig. 3.
§ 33. Tt seems not improbable that if the figure deviates by ever so little from being exactly ellipsoidal, Maxwell’s condition might be fulfilled. It seems indeed quite probable that Maxwell's condition (see §§ 13, 29, above) is fulfilled by a geodetic on a closed surface of any shape in general, and that exceptional cases, in which the question of §29 is to be answered in the negative, are merely particular surfaces of definite shapes, infinitesimal deviations from which will allow the question to be answered in the affirmative.
§ 34. Now with an affirmative answer to the question—is Maxw'ell's condition fulfilled ?—what does the Boltzmann-Maxwell doctrine assert in respect to a geodetic on a closed surface ? The mere wording of Maxwell's statement, quoted in §13 above, is not applicable to this case, but the meaning