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HISTORY OF PHYSICAL ASTRONOMY.

the mean distances were concerned, the mean motions of the planets wereinvariable by the second as well as by the first approximation.

The interesting result to which Poisson was conducted by his researches,had the effect of again inducing Lagrange and Laplace to direct their at-tention to the theory of planetary perturbation. Poisson , after the ex-ample of these geometers, had assumed, as the basis of his investigation,the elliptic equations of the undisturbed orbit, and obtained the variationsof the elements, by supposing the constants in these equations to be everyinstant changing their values by infinitely small quantities. Lagrange,however, was led, by reflecting on Poisson ’s researches, to consider the sub-ject under a more general aspect. He assumed as the basis of his inves-tigation the three differential equations of the planet’s motion, derivedfrom the supposition that it was exposed only to the central action of thesun. Each of these equations being of the second order, its complete in-tegral would contain two arbitrary constants, and, therefore, the three equa-tions which determine the place of the planet in its orbit would containsix arbitrary constants-. When the action of the planets is taken into ac-count, another quantity is introduced into the differential equations, termedthe perturbing function. Conceiving, then, this function to arise from acontinuous variation of the six constants in the primitive equations, La-grange succeeded in obtaining the expressions for the variation of eachconstant, without assuming the ellipticity of these equations. The varia-tions, when investigated by this uniform process, exhibited a very remark-able form, being all expressed by partial differentials of the perturbingfunction, taken with respect to the constants, and multiplied into quan-tities, which were functions only of the constants. This form presented agreat advantage in practice, for, as it was possible to expand the perturbingfunction into a series of sines and cosines of angles, increasing propor-tionally with the time, the secular variations of the elements were at onceobtained by means of the terms that were explicitly independent of thetime, while the remaining terms being integrated, and then substituted inthe formulae for finding the place of a planet in an elliptic orbit, gave thelongitude and latitude of the planet in the disturbed orbit correspondingto an}' assignable time. Lagrange then applied his analysis to the ques-tion of the invariability of the mean distance. Poisson had not consideredthe effect of the variations of the elements of the disturbing planets, onaccount of the analytical difficulties offered by the perturbing function,which did not preserve the same form throughout the investigation whenthat supposition was introduced. Lagrange got rid of this inconvenienceby transferring the origin of co-ordinates from the centre of the sun tothe centre of gravity of the sun and planets. The perturbing functionbeing then symmetrical with respect to all the planets, the same analysiswas applicable, whatever might be the planet whose elements were sup-posed to vary. Lagrange by this means succeeded in shewing that thesecond approximation would not introduce into the expression for themean distance any term which increased constantly with the time, evenwhen the variations of the elements of the disturbing planets were in-cluded in the investigation.

It is right to mention that the variations of most of the elementshad already appeared in the form under which Lagrange just presentedthem. That great geometer had so expressed the variation of the mean

* Memoires de l’lnstitut, 1808.