MOTION OF THE LUNAK APSIDES.

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to revolve in the direction EADB. Then, by drawing the figurein various situations of P throughout the whole circle, the readerwill easily satisfy himself—1st, That the tangential force accele-rates P, as it moves from E towards A, and from D towards B,but retards it as it passes from A to D, and from B to E. 2nd,That the tangential force vanishes at the four points A, D, E, B,and attains a maximum at some intermediate points. 3rdly, Thatthe normal force is directed outwards at the syzygies A, B, andinwards at the points D, E, at which points respectively its out-ward and inward intensities attain their maxima. Lastly, that thisforce vanishes at points intermediate between A D, D B, B E, andE A, which points, when M is considerably remote, are situatednearer to the quadrature than the syzygies.

(676.) In the lunar theory, to which we shall now proceed toapply these principles, both the geometrical representation and thealgebraic expression of the disturbing forces admit of great sim-plification. Owing to the great distance of the sun M, at whosecentre the radius of the moon’s orbit never subtends an angle ofmore than about 8', N P may be regarded as parallel to A B.And D S E becomes a straight line coincident with the line ofquadratures, so that V P becomes the cosine of A S P to radius

Fig. 90.

S P, and N L=N P . sin A S P; L P=N P. cos A S P. More-over, in this case (see the note on the last article) N P=3 P V=3 S P . cos ASP; and consequently N L=3 S P . cos A S P. sinA S P=f S P . sin 2 A S P, and L P=S P (3. cos A S P 2 —1)=J S P (1 +3 . cos 2 A S P) which vanishes when cos A S P 2 =J,or at 64° 14’ from the syzygy. Suppose through every point ofof P’s orbit there be drawn S Q=3 S P . cos A S P 2 , then will Q