Page:The Solar System - Six Lectures - Lowell.djvu/95



and the term has a large coefficient, and therefore a large value.

If, then, the mean motions, and therefore the periods, of perturber and perturbed are commensurable, the disturbing effect upon the major axes of each will be great. The major axes will be altered until the periods cease to be commensurable, and it will be long before perturbation brings them back to commensurability again.

Furthermore, the least value $$\textstyle l+m$$can have is $$\textstyle p-q$$, while the period of the action of the term is $$\textstyle \frac{2\pi}{pn-qn'};$$; whence the greatest term is when $$p$$ and $$q$$ are both as small as possible, since conjunctions will occur oftener in proportion as $$q$$ is small.

Geometrically, the effect can be seen in the following way. Clearly, the disturbing pull is considered, greatest when the two bodies are in conjunction, and so long as the periods are incommensurable, conjunctions will occur in different parts of the orbit successively, and thus neutralize one another's effect upon the major axes. But if the periods of the two bodies be commensurable, conjunctions will occur in the same place over and over again, and the major axes will be altered there without compensatory alterations elsewhere;and this will go on until the major axes are so