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three he in the position of full moon at the same time. Con-sequently, only two of them can he eclipsed at once. Themotion of the 4th is not connected in this way with that ofthe other three. It may therefore he eclipsed with two of theothers ; in which case Jupiter would have only one moonvisible. And even that might he in the phase of new moon;in fact, if two of the three eclipsed were the 2nd and 3rd moons,the same relation previously named would require that theone not eclipsed (Le., the 1st) must he a new moon; underwhich circumstances the Jovian skies would for the time bedeprived of all moonlight.
The relation to which we refer was discovered by Laplace.It may be stated somewhat technically as follows :—That themean angular motion of the 1st Satellite round Jupiter , addedto twice that of the 3rd, equals three times that of the 2nd;while it is also the case that, if the 2nd and 3rd are, at anygiven time, exactly in the same direction in longitude, or(neglecting the inclination of their orbits) exactly in the samestraight line as seen from Jupiter , the 1st must he exactly inthe opposite direction.
If therefore, as we stated above, the 2nd and 3rd he supposedto he together in the position of full moon, the 1st mustnecessarily, by the second part of Laplace’s theorem, he inthat of new moon ; while a little consideration of the formerpart of the theorem in question will show, that the three cannever be full moons, or new moons, or in fact be all seen inany given direction, at the same time.
It will be remembered that we stated (see page 329) that theangular movement of the 2nd is not far from being equal totwice that of the 3rd, while that of the 1st is almost exactlydouble that of the 2nd; their angular velocities being just somuch greater as their periodic times are shorter. The num-bers 1, 2, and 4 therefore very nearly represent the ratios oftheir respective velocities. If they did so precisely, it is ofcourse evident, since 4 plus twice 1 equals three times 2, thatthe first part of Laplace’s law of their motions would hesatisfied.
But that law, as discovered by Laplace, involves the truth