(2)
From these may be derived by differentiation as to t the velocities
| dx | = ƒ′1(a, b, c, d, t) = x′ |
| dt |
| dy | = ƒ′2(a, b, c, d, t) = y′ |
| dt |
(3)
The symbols x′ and y′ are used for brevity to mean the velocities expressed by the differential coefficients. The arbitrary constants, a, b, c and d, are the elements of the orbit, or any quantities from which these elements can be obtained. We note that, in the actual process of integration, no geometric construction need enter.
| Fig. 2. |
Let us next consider the problem in another form. Conceive that instead of the orbit of the planet, there is given a position P (fig. 2), through which the planet passed at an assigned moment, with a given velocity, and in a given direction, represented by the arrowhead. Logically these data completely determine the orbit in which the planet shall move, because there is only one such orbit passing through P, a planet moving in which would have the given speed. It follows that the elements of the orbit admit of determination when the co-ordinates of the planet at an assigned moment and their derivatives as to time are given. Analytically the elements are determined from these data by solving the four equations just given, regarding a, b, c and d as unknown quantities, and x, y, x′, y′ and t as given quantities. The solution of these equations would lead to expressions of the form
a = φ1(x, y, x′, y′, t)
b = φ2(x, y, x′, y′, t)
&c. &c.
(4)