All the other superior planets are so far away from the sun and earth, that the sides which they turn towards the sun and the earth in every part of their orbit are so nearly the same, that no change in the form of their disks can be detected. 135. _The Synodical Period of a Superior Planet._--During a synodical period of a superior planet the earth must gain one revolution, or 360°, on the planet, as will be evident from an examination of Fig. 152, in which _S_ represents the sun, _E_ the earth, and _P_ the planet at opposition. Before the planet can be in opposition again, the earth must make a complete revolution, and overtake the planet, which has in the mean time passed on from _P_ to _P'_. [Illustration: Fig. 152.] In the case of most of the superior planets the synodical period is shorter than the sidereal period; but in the case of Mars it is longer, since Mars makes more than a complete revolution before the earth overtakes it. The synodical period of a superior planet is found by direct observation. 136. _The Sidereal Period of a Superior Planet._--The sidereal period of a superior planet is found by a method of computation similar to that for finding the sidereal period of an inferior planet:-- Let _a_ denote the synodical period of the planet, Let _b_ denote the sidereal period of the earth, Let _x_ denote the sidereal period of the planet. Then will _360°/b_ = daily motion of the earth, And _360°/x_ = daily motion of the planet; Also _360°/b - 360°/x_ = daily gain of the earth. But _360°/a_ = daily gain of the earth: Hence _360°/b - 360°/x = 360°/a_ _1/b - 1/x = 1/a_ _ax - ab = bx_ _(a-b)x = ab_ _x = ab/(a-b)_. [Illustration: Fig. 153.] 137. _The Relative Distance of a Superior Planet._--Let _S_, _e_, and _m_, in Fig. 153, represent the relative positions of the sun, the earth, and Mars, when the latter planet is in opposition. Let _E_ and _M_ represent the relative positions of the earth and Mars the day after opposition. At the first observation Mars will be seen in the direction _emA_, and at the second observation in the direction _EMA_.