504

APPENDIX.

tangent. Hence the upper focus of the new ellipse must be somewhere I

in the line p h'. Again, by another property of the ellipse, the sum of the j

same two lines is equal to the major axis. Hence s and h' will represent I

the foci of an ellipse whose tangential angle at p is equal to s p x, and j

whose major axis is equal to that of the undisturbed orbit. But by the j

principles of dynamics, the major axis of the ellipse is not altered by 1the disturbing force. Hence the truth of the proposition is manifest. jBy varying the position of p, and supposing the disturbing force as ;tending either to increase or diminish the attractive force at s, the jdifferent results referred to in (92), (23), and (24) may be very easily ;deduced. j

(38.) Next let the disturbing force act in the direction of the tangent, j

so as to retard or accelerate the velocity of the planet. Let it be sup- j

posed to increase the velocity and let p be the point at which it acts. In

this case the tangential angle at p is not ]

affected by the disturbing force, but the major :

axis of the ellipse is increased by its action.Produce ph so that h h' may represent theincrement of the major axis occasioned by thedisturbing force. Join s h', and bisect theline s h in c'. Then will s o' represent thenew eccentricity of the orbit, and a's h' b'the new position of the line of apsides. The '

truth of this proposition is so obvious as torender any formal demonstration of it super-fluous. The various theorems announced in (30), (31), and 32) are easilydeducible from it.

II.

APPLICATION OF THE FOREGOING PKINCIPLES TO CERTAIN CASES OF ACTUAL

PERTURBATION.

(39). Let us suppose two comparatively small bodies to be revolving in ;circular orbits situate in the same plane, round a large central body s, and let jthe mean motion of the interior revolving body be almost exactly double the ]mean motion of the exterior one. Let us assume also, for facility of explana- jtion, that the exterior body maintains a fixed position at p, while the interior I

body performs an entire revolution around )s. Join s p by a straight line, cutting theorbit of the interior body in a and d. iThen dap will represent the line of ■conjunction of the two bodies. Now in !those cases of the solar system where- ;in the mean motion of one revolving 1body is almost exactly double the mean jmotion of the other, the effects producedby the mutual perturbation of the twobodies are sensible only near conjunc-tion. Let us suppose that in one ofsuch cases the disturbing influence of the exterior body first becomes sen-sible when the interior body has arrived at b, a position somewhat less ad-vanced than the line of conjunction. It may be easily shewn that the disturb-

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