Now let us see how we can use the quantities _r_, θ, _i_, and Ω as well as Kepler’s two time laws to determine the motion of the satellite in space. Suppose that we have made observations of the Telstar at two times _t_₁ and _t_₂ and that we have measured its distance along lines ρ₁ and ρ₂ in _Figure 7_. In other words, we know that at these two times the satellite was at the points P₁ and P₂. Since three points determine a plane, we know in this case that P₁, P₂, and O define the satellite orbital plane. Knowing this, we can now calculate the angles θ₁ and θ₂, the distances _r_₁ and _r_₂, and the angles _i_ and Ω. (The detailed formulas for this are derived from analytic geometry.) [Illustration: _Figure 7_] However, we still do not know the length and the width of the particular ellipse the satellite is following and how this orbit is oriented within the orbital plane. Let us imagine again that we can stand off to one side of the orbit and take a good look at it; _Figure 8_ shows us what we would see. There are the two points P₂ and P₁ at which we have observed the satellite. We know the positions of these points relative to each other and in relation to the center of the earth, because we have already calculated _r_₂, _r_₁, θ₂, and θ₁, But any number of ellipses could be made to pass through these two points. Some might be very large, others might be so narrow that they would intersect the earth and thus be impossible. However, only one of these ellipses will satisfy the time difference that we observed between P₁ and P₂. In other words, the shape and period of this particular ellipse must be such that it will cause the satellite to pass through P₁ and P₂ in exactly the time interval _t_₂ - _t_₁. If we work out our time formulas, we will convince ourselves that there is only one such ellipse. When we have found it, we have determined the orbit of the Telstar satellite from the two observed positions and times. [Illustration: _Figure 8_] [Illustration: _Figure 9_]