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 _360°/b_ = the daily motion of the earth, And _360°/x_ = the daily motion of the planet, And _360°/x - 360°/b_ = the daily gain of the planet: Also _360°/a_ = the daily gain of the planet: Hence _360°/x - 360°/b = 360°/a_. Dividing by 360°, we have _1/x - 1/b = 1/a_; Clearing of fractions, we have _ab - ax = bx_: Transposing and collecting, we have _(a + b)x = ab_: Therefore _x = ab/a+b_. 130. _The Relative Distance of an Inferior Planet._--By the _relative distance_ of a planet, we mean its distance from the sun compared with the earth's distance from the sun. The relative distance of an inferior planet may be found by the following method:-- [Illustration: Fig. 148.] Let _V_, in Fig. 148, represent the position of Venus at its greatest elongation from the sun, _S_ the position of the sun, and _E_ that of the earth. The line _EV_ will evidently be tangent to a circle described about the sun with a radius equal to the distance of Venus from the sun at the time of this greatest elongation. Draw the radius _SV_ and the line _SE_. Since _SV_ is a radius, the angle at _V_ is a right angle. The angle at _E_ is known by measurement, and the angle at _S_ is equal to 90°- the angle _E_. In the right-angled triangle _EVS_, we then know the three angles, and we wish to find the ratio of the side _SV_ to the side _SE_. The ratio of these lines may be found by trigonometrical computation as follows:-- _VS : ES = sin SEV : 1._ Substitute the value of the sine of SEV, and we have _VS : ES = .723 : 1._ Hence the relative distances of Venus and of the earth from the sun are .723 and 1. Superior Planets. 131. _The Superior Planets._--The _superior planets_ are those which lie beyond the earth. They are _Mars_, the _Asteroids_, _Jupiter_, _Saturn_, _Uranus_, and _Neptune_. [Illustration: Fig. 149.]