The perturbations of the planets is the subject of reiterated researches by all the great geometers who have raised up Physical Astronomy to its present elevation. They have been successful in determining the variations which the elements of the orbit of a disturbed planet undergo; and in expressing these variations analytically, in the manner best adapted for computation. But the inquirer who turns his attention to this branch of study will find that it is made to depend upon a theory in mechanics, which is one of considerable analytical intricacy, known by the name of the Variation of the Arbitrary Constants. Considerations similar to those employed in this theory were found necessary in Physical Astronomy from its origin; but the genius of Lagrange imagined and completed the analytical processes of general application. In a dynamical problem which is capable of an exact solution, such as a planet revolving by the central attraction of the sun, the formulas constructed by Lagrange enable us to ascertain the alterations that will be induced on the original motions of the body, if we suppose it urged by new and very small forces, such as the irregular attractions of the other bodies of the planetary system. General views of this nature are very valuable, and contribute greatly to the advancement of science. But their application is sometimes attended with inconvenience. In particular cases, the general structure of the formulas may require a long train of calculation, in order to extricate the values of the quantities sought. It may be necessary for attaining this end to pass through many differential equations, and to submit to much subordinate calculation. The remedy for this inconvenience seems to lie in separating the general principles from the analytical processes by which they are carried into effect. In some important problems, a great advantage, both in brevity and clearness, will be obtained by adapting the investigation to the particular circumstance of the case, and attending solely to the principles of the method in deducing the solution. It may therefore become a question whether it be not possible to simplify physical astronomy by calling in the aid only of the usual principles of dynamics, and by setting aside every formula or equation not absolutely necessary for arriving at the final results. The utility of such an attempt, if successful, can hardly be doubted. By rendering more accessible a subject of great interest and importance, the study of English mathematicians may be recalled to a theory which, although it originated in England, has not received the attention it deserves, and which it
has
met with in foreign countries. The paper which I have the honour to submit to the Royal Society, contains a complete determination of the variable elements of the elliptic orbit of a disturbed planet, deduced from three differential equations that follow readily from the mechanical conditions of the problem. In applying these equations, the procedure is the same whether a planet is urged by the sole action of the central force of the sun, or is besides disturbed by the attraction of other bodies revolving about that luminary; the only difference being that, in the first case, the elements of the orbit are all constant, whereas in the other case they are all variable. The success of the method here followed is derived from a new differential equation between the time and the area described by the planet in its momentary plane, which greatly shortens the investigation by making it unnecessary to consider the projection of the orbit. But the solution in this paper, although no reference is made to the analytical formulas of the theory of the variation of the arbitrary constants, is no less an application of that method, and an example of its utility and of the necessity of employing it in very complicated problems.
Residual Elastic Strains in Autofrettaged Tubes: Elastic–Ideally Plastic Model Analysis